Warwick Mathematics Institute Events

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Upcoming Seminars

Combinatorics on 17 March 2023 at 14:00 in B3.02

Speaker: Joseph Hyde (UVic)
Title: Turan Colourings in Off-Diagonal Ramsey Multiplicity

Abstract: The Ramsey multiplicity problem asks for the minimum asymptotic density of monochromatic labelled copies of a graph $H$ in a red/blue colouring of $K_n$. As with explicitly calculating Ramsey numbers, this problem is notoriously difficult, with it being solved for only a handful of graphs $H$. Here, and in an accompanying paper ('Off-Diagonal Ramsey Multiplicity' by Jonathan Noel and Elena Moss), we introduce an off-diagonal generalization in which the goal is to minimize a certain naturally weighted sum of the densities of red copies of one graph and blue copies of another.

In this talk we will focus on when colourings based on the Tur\'{a}n-graph appear as such minimizers, in particular proving a result based on a recent breakthrough of Fox and Wigderson, who answered the Ramsey multiplicity problem for a certain class of uncommon graphs. Joint work with Jonathan Noel and Jae-baek Lee.

Colloquium on 17 March 2023 at 16:00 in B3.02

Speaker: Catherine Powell (Manchester)
Title: The How and Why of Being Intrusive

Abstract: In engineering applications (heat transfer, fluid flow, elasticity etc), we often encounter physics-based models consisting of partial differential equations (PDEs) with uncertain inputs, which are reformulated as so-called parametric PDEs. Given a probability distribution for the inputs, the forward UQ problem consists of trying to estimate statistical quantities of interest related to the model solution. Conversely, given (usually, noisy) data relating to the model solution, the inverse UQ problem consists of trying to infer the uncertain inputs themselves.

Over the last two decades a myriad of numerical schemes have been developed to tackle the forward and inverse UQ problems for PDE models. The vast majority of these are sampling schemes and are non-intrusive in the sense that users do not have to modify existing solvers and codes for the associated deterministic PDEs. This is very attractive in industrial settings. Stochastic Galerkin methods, also known as intrusive polynomial chaos methods, standard apart and are much less widely used in practise. In this talk, I will outline the intrusive approach, its advantages and limitations, and explain why, if properly implemented, it sometimes offers advantages over sampling methods.

Past Seminars

AGATA on 16 March 2023 at 17:00

Speaker: Javier Aguilar Martin (Univeristy of Kent)
Title: Introduction to derived A-infinity-algebras

Abstract: We introduce the notion of homotopy associative (A-infinity-) algebras in the framework of operads, with loop spaces as our main example. We will motivate the study of these algebraic structures through the theory of minimal models. We will also see their interaction with brace algebras coming from operadic structures. Derived A-infinity-algebras will be introduced to overcome the limitations of classical A-infinity-algebras.

Probability Theory on 15 March 2023 at 16:00

Speaker: Neil O'Connell (University College Dublin)
Title: A (Toda-lly cool) Markov chain on reverse plane partitions

Abstract: Abstract: I will discuss a natural Markov chain on reverse plane partitions which is closely related to the Toda lattice and has some remarkable properties. This talk is based on the paper https://arxiv.org/abs/2211.05718 and aimed at a general probability audience. (The fun part of the title is borrowed from `Toda-lly cool stuff’, an informal and very accessible introduction to the Toda lattice, by Barbara Shipman, available at https://faculty.tcu.edu/richardson/seminars/barbaratoda.pdf.)

Algebraic Topology on 14 March 2023 at 16:00

Speaker: Jeffrey Carlson (Imperial College)
Title: Products on Tor and strong homotopy commutativity

Abstract: The Eilenberg–Moore spectral sequence converges from the classical Tor of a span of cohomology rings to the differential Tor of a span of cochain algebras (which is the cohomology of the homotopy pullback). These are both rings, for very different reasons: the first structure comes about because cohomology rings are commutative, and the second arises as a corollary of the Eilenberg—Zilber theorem.

One might well ask when a more general differential Tor of DGAs admits a ring structure, though apparently no one did. We will show that when the DGAs in question admit a certain sort of $E_3$-algebra structure generalizing the previous examples, Tor is a commutative graded algebra.

We have not done this out of an innocent interest in homotopy-commutative algebras. In the 1960s and '70s there was a flurry of activity developing A-infinity-algebraic techniques with an aim toward computing the Eilenberg–Moore spectral sequence (for example, of a loop space or homogeneous space). Arguably the most powerful result this program produced was the 1974 theorem of Munkholm that the sequence collapses when the three input spaces have polynomial cohomology over a given principal ideal domain, which however only gives the story on cohomology groups. Our result shows that Munkholm's map is in fact an isomorphism of rings.

This work is joint with several large commutative diagrams.

Algebra on 13 March 2023 at 17:00

Speaker: Vanthana Ganeshalingam (University of Warwick)
Title: Subgroup Structure of Reductive Groups

Abstract: This talk will introduce the concept of G complete-reducibility (G c-r) originally thought of by Serre in the 90s. This idea has important connections to the open problem of classifying the subgroups of a reductive group G. I will explain the methodology of the classification so far and the main obstacle which is understanding the non-G-cr subgroups.

Number Theory on 13 March 2023 at 15:00

Speaker: Cecile Dartyge (Lorraine)
Title: On the largest prime factor of quartic polynomial values : the cyclic and dihedral cases.

Abstract: Let $P(X)$ be an irreducible, monic, quartic polynomial with integral coefficients and with cyclic or dihedral Galois group.
There exists c_P >0 such that for a positive proportion of integers n, P(n) has a prime factor bigger than n^{1+c_p}.
This is a joint work with James Maynard.

Colloquium on 10 March 2023 at 16:00

Speaker: Matej Balog (DeepMind London)
Title: Discovering faster matrix multiplication algorithms with reinforcement learning

Abstract: Improving the efficiency of algorithms for fundamental computational tasks such as matrix multiplication can have widespread impact, as it affects the overall speed of a large amount of computations. Automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. In this talk I’ll present AlphaTensor, our learned agent for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time since its discovery 50 years ago. In this talk I’ll present the mathematical problem underlying algorithm discovery for matrix multiplication, and our formulation of this problem as a single-player game. Then I'll describe general ingredients for tackling mathematical problems using machine learning techniques, and show how these ingredients enable AlphaTensor to learn to play this challenging game well, and thereby to discover faster algorithms for matrix multiplication.

Combinatorics on 10 March 2023 at 14:00

Speaker: Jane Tan (Oxford)
Title: Reconstructing 3D cube complexes from boundary distances

Abstract: TBA

AGATA on 09 March 2023 at 17:00

Speaker: James Timmins (University of Oxford)
Title: Dimensions in noncommutative algebra

Abstract: Due to the dictionary between commutative algebra and algebraic geometry, the
size of commutative algebras and their modules can be measured by the
dimension of geometric objects. In this talk, I’ll describe the picture for
noncommutative algebra, focusing on a fundamental invariant known as
canonical dimension. I’ll illustrate the theory with a classical example that leads
to Bernstein’s inequality, and then describe recent progress for completed group
algebras of p-adic groups, giving a connection to the (mod p) Langlands
programme.

Geometry and Topology on 09 March 2023 at 15:00

Speaker: Nansen Petrosyan (Univeristy of Southampton)
Title: Hyperbolicity and L-infinity cohomology

Abstract: L-infinity cohomology is a quasi-isometry invariant of finitely generated groups.
It was introduced by Gersten as a tool to find lower bounds for the Dehn
function of some finitely presented groups. I will discuss a generalisation of a
theorem of Gersten on surjectivity of the restriction map in L-infinity
cohomology of groups. This leads to applications on subgroups of hyperbolic
groups, quasi-isometric distinction of finitely generated groups and L-infinity
cohomology calculations for some well-known classes of groups such as RAAGs,
Bestvina-Brady groups and Out(F_n). Along the way, we obtain hyperbolicity
criteria for groups of type FP_2(Q) and for those satisfying a rational homological
linear isoperimetric inequality.
I will first define L-infinity cohomology and discuss some of its properties. I will
then sketch some of the main ideas behind the proofs. This is joint work with
Vladimir Vankov.

Geometry and Topology on 09 March 2023 at 14:00

Speaker: Elia Fioravanti (MPIM Bonn)
Title: Coarse cubical rigidity.

Abstract: When a group G admits nice actions on CAT(0) cube complexes, understanding
the space of all such actions can provide useful information on the outer
automorphism group Out(G). As a classical example, the Culler-Vogtmann outer
space is (roughly) the space of all geometric actions of the free group F_n on a 1-
dimensional cube complex (a tree). In general, however, spaces of cubulations
tend to be awkwardly vast, even for otherwise rigid groups such as the hexagon
RAAG. In an attempt to tame these spaces, we show that all cubulations of many
right-angled Artin and Coxeter groups coarsely look the same, in a strong sense:
they all induce the same coarse median structure on the group. Joint work with
Ivan Levcovitz and Michah Sageev.

Probability Theory on 08 March 2023 at 16:00

Speaker: Natasha Blitvic (Queen Mary University London)
Title: Combinatorial moment sequences

Abstract: Take your favourite integer sequence. Is this sequence a sequence of moments of some probability measure on the real line? We will look at a number of interesting examples (some proven, others merely conjectured) of moment sequences in combinatorics. We will consider ways in which this positivity may be expected (or surprising!), the methods of proving it, and the consequences of having it. The problems we will consider will be very simple to formulate, but will take us up to the very edge of current knowledge in combinatorics, ‘classical’ probability, and noncommutative probability.

Algebraic Geometry on 08 March 2023 at 15:00

Speaker: Farhad Babaee (Bristol)
Title: Some applications of Tropical Geometry in Complex Geometry

Abstract: In this talk, I will recall two important questions in Complex Analytic Geometry, namely, a strong version of the Hodge Conjecture for Positive Currents and the Equidistribution Conjecture of Dinh--Sibony. I will also explain how Tropical Geometry can provide insight into these questions. No background in these topics is assumed.

Algebraic Topology on 07 March 2023 at 16:00

Speaker: Ben Briggs (University of Copenhagen)
Title: Syzygies of the cotangent complex

Abstract: The cotangent complex is an important object from commutative algebra. It was defined by Quillen using homotopical methods, and is usually usually extremely difficult to compute. It is connected with some more tractable invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies of the cotangent complex. One can try to establish higher analogues of the Jacobian criterion by characterising geometric conditions in terms of homological properties of these syzygies. I will explain how thinking along these lines leads to a new proof of Quillen's conjecture on the cotangent complex and Vasconcelos' conjecture on the conormal module. I'll also try to explain some of the parallels in rational homotopy theory. This is joint work with Srikanth Iyengar.

Number Theory on 06 March 2023 at 15:00

Speaker: Peiyi Cui (East Anglia)
Title: Decompositions of the category of l-modular representations of SL_n(F)

Abstract: Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with respect to several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

Colloquium on 03 March 2023 at 16:00

Speaker: Olivia Caramello (University of Insubria, Como; online)
Title: Grothendieck toposes as unifying 'bridges' in Mathematics

Abstract: I will explain the sense in which Grothendieck toposes can act as unifying 'bridges' for relating different mathematical theories to each other and studying them from a multiplicity of points of view. I shall first present the general techniques underpinning this theory and then discuss a number of selected applications in different mathematical fields.

Combinatorics on 03 March 2023 at 14:00

Speaker: Jozef Skokan (LSE)
Title: Separating the edges of a graph by a linear number of paths

Abstract: Recently, Letzter proved that any graph of order n contains a collection P of O(n log^*n) paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f. We improve this upper bound to 19n, thus answering a question of Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhar and by Falgas-Ravry, Kittipassorn, Korandi, Letzter, and Narayanan.

Applied Mathematics on 03 March 2023 at 12:00

Speaker: Carina Geldhauser (Lund)
Title: An introduction to point vortex models

Abstract: In this talk we will discuss a family of discrete models for atmospheric turbulence, often called point vortex models. They have been originally derived by Helmholtz, about 130 years ago, but many interesting questions are still open.

We will show how point vortices provide an approximation of solutions to generalized surface quasi-geostrophic models, a family of fractional PDEs which interpolate between 2D Euler equations and the more irregular SQG equation.

Lastly, we will briefly touch upon how variational methods and tools from probability theory together can help us to obtain more information about turbulence phenomena. Joint work with Marco Romito (Uni Pisa).

Junior Analysis and Probability Seminar on 02 March 2023 at 13:00

Speaker: Martin Peev (Imperial College London)
Title: Localising Fermionic (S)PDEs

Abstract: TBA

Probability Theory on 01 March 2023 at 16:00

Speaker: Serge Cohen (University of Toulouse)
Title: Transition of the simple random walk on the graph of the ice-model

Abstract: The 6-vertex model is a seminal model for many domains in Mathematics and Physics. The sets of configurations of the 6-vertex model can be described as the set of paths in multigraphs. In this article the transition probability of the simple random walk on the multigraphs is computed. The unexpected point of the results is the use of continuous fractions to compute the transition probability.

Algebraic Geometry on 01 March 2023 at 15:00

Speaker: Liana Heuberger (Bath)
Title: Q-Fano threefolds and how to construct them

Abstract: I will give a brief overview about the techniques involved in constructing Fano varieties using mirror symmetry, appearing in the works of Coates, Corti, Kasprzyk et al. I will then describe one of its more concrete incarnations, a method of "inverting" a toric degeneration called Laurent Inversion, which I have used to construct 100 deformation families of Q-Fano threefolds.

Postgraduate on 01 March 2023 at 12:00

Speaker: Patricia Medina Capilla (University of Warwick)
Title: Crowns and their uses in generation problems

Abstract: The generation properties of a group reveal a lot of information about its structure. As such, these properties have been investigated very thoroughly in the last century. This has led to some very beautiful and surprising results being proven, such as the fact that all simple groups are generated by two elements. Relatively recently, Dalla Volta and Lucchini developed the theory of crowns in order to tackle such problems.

In this talk, we will introduce the set of crowns of a group, describe how it is utilised in generation problems, and showcase its strength via some examples. Time allowing, I will explain how this approach has been applied in recent research in order to bound the number of generators of the maximal subgroups of simple groups.

Algebraic Topology on 28 February 2023 at 16:00

Speaker: Gonçalo Tabuada (Warwick)
Title: Grothendieck classes of quadric hypersurfaces and involution varieties

Abstract: The Grothendieck ring of varieties, introduced in a letter from Alexander Grothendieck to Jean-Pierre Serre (August 16th 1964), plays an important role in algebraic geometry. However, despite the efforts of several mathematicians, the structure of this ring still remains poorly understood. In this talk, in order to better understand the Grothendieck ring of varieties, I will describe some new structural properties of the Grothendieck classes of quadric hypersurfaces and involution varieties. More specifically, by combining the recent theory of noncommutative motives with the classical theory of motives, I will show that if two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, this implies in numerous cases (e.g., when the base field is a local or global field) that two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic.

Number Theory on 27 February 2023 at 15:00

Speaker: Efthymios Sofos (Glasgow)
Title: Schinzel's Hypothesis on average and the Hasse principle

Abstract: Schinzel's Hypothesis regards prime values taken by a polynomial with integer coefficients. I will talk about work with Skorobogatov where we established that the Hypothesis holds for 100% of cases in a probabilistic sense. I will also talk about joint work with Browning and Teräväinen where we extend the previous result in various directions.

Colloquium on 24 February 2023 at 16:00

Speaker: Steven Tobias (Leeds)
Title: "So Many DynamoS": Some interesting mathematical problems in dynamo theory

Abstract: The generation of magnetic field in the Earth's interior and the origin of the eleven year solar cycle are both thought to lie in hydromagnetic dynamo action. In both cases fluid motions interact with rotation to sustain electrical currents and hence magnetic field. In this talk I will give an introduction to the mathematical structure of the equations for the generation of magnetic field and highlight some interesting unsolved mathematical problems. I'll conclude by drawing an analogy with transition to turbulence in flow down a pipe.

Combinatorics on 24 February 2023 at 14:00

Speaker: Amarja Kathapurkar (University of Birmingham)
Title: Transversal cycle factors in multipartite graphs

Abstract: TBA

Applied Mathematics on 24 February 2023 at 12:00

Speaker: Ellen Luckins (Oxford)
Title: Reactive decontamination of porous media

Abstract: Following a chemical weapons attack, it is crucial for public health that the toxic chemical agent is properly cleaned up. One particular issue is when the agent has contaminated porous materials, such as brick or concrete. In such cases, decontamination is typically achieved by neutralising the agent with a cleanser in a chemical reaction. It is relatively straightforward to write down a model that describes the interplay of the agent and cleanser fluids on the scale of the pores, but very computationally expensive to solve such a model over realistic spill sizes. In this talk I will present homogenised PDE models for the reactive decontamination of porous media, which are computationally efficient to simulate while still taking the pore-scale behaviour into account. Solutions of these homogenised models show how differences in the initial distribution of agent within the pore-space affect both the decontamination time and the amount of cleanser required to fully decontaminate the porous material.

Junior Analysis and Probability Seminar on 23 February 2023 at 13:00

Speaker: Dimitri Bytchenkoff (University of Vienna)
Title: Frames and kernel theorems for co-orbit spaces

Abstract: TBA

Probability Theory on 22 February 2023 at 16:00

Speaker: Alessandra Occelli (Université d'Angers)
Title: Nonlinear fluctuations of multi-species interacting particle sys

Abstract:
Abstract:
We study the equilibrium fluctuations of an exclusion process evolving on the discrete ring with three species of particles, named A, B and C . We prove that proper choices of the density fluctuation fields (given by linear combinations of the fields associated to the conserved quantities that match the prediction from mode coupling theory [Spohn 2014]) converge, in a suitable large scale limit, to stochastic partial differential equations, that can either be the Ornstein--Uhlenbeck equation or the stochastic Burgers equation. Based on a joint work with G. Cannizzaro, P. Gonçalves and R. Misturini.

Postgraduate on 22 February 2023 at 12:00

Speaker: Hefin Lambley (University of Warwick)
Title: An introduction to inverse problems

Abstract: Given an observed effect, the inverse problem is to determine the cause. These problems are hard to solve because they are unstable: small errors in the observed effect lead to big errors in the reconstructed cause.
I will give some examples of inverse problems and discuss practical ways to solve them. We'll also see how this links to uncertainty quantification, which is an active research topic both for applied mathematicians and for engineers modelling real-world phenomena in the presence of noise and uncertainty.

Algebra on 20 February 2023 at 17:00

Speaker: Nadia Mazza (Lancaster)
Title: On endotrivial modules

Abstract: Let G be a finite group and k a field of positive characteristic p diving the order of G. An endotrivial kG-module is a finitely generated kG-module which is "invertible" in some suitable sense. Since the late 70s, these modules have been intensely studied in modular representation theory. In this talk, we briefly review the background, before presenting some results (joint with Carlson, Grodal and Nakano) about endotrivial modules for some "very important" finite groups.

Number Theory on 20 February 2023 at 15:00

Speaker: Sacha Mangerel ((Durham))
Title: (Durham)

Abstract: A conjecture of Chowla, analogising the Hardy-Littlewood prime $k$-tuples conjecture, predicts that the autocorrelations of $\lambda$ (the completely multiplicative function taking the value -1 at all primes) tend to 0 on average, e.g., $\frac{1}{x}\sum_{n \leq x} \lambda(n+1)\cdots \lambda(n+k) \rightarrow 0 \text{ as } x \rightarrow \infty$.

This conjecture, along with its generalisation to other bounded ``non-pretentious'' multiplicative functions due to Elliott, remain wide open for $k \geq 2$. In this talk I will present an explicit construction of a non-pretentious multiplicative function $f: \mathbb{N} \rightarrow \{-1,1\}$ all of whose auto-correlations tend to 0 on average, answering a question of Lemanczyk. I will further discuss the following applications of this construction:

1. a proof that Chowla's conjecture does not imply the Riemann Hypothesis, i.e., there are functions $f$ all of whose autocorrelations tend to 0, but that do not exhibit square-root cancellation on average (the object of some recent speculation);

2. there are multiplicative subsemigroups of $\mathbb{N}$ with Poissonian gap statistics, thus giving an unconditional multiplicative analogue of a classical result of Gallagher about primes in short intervals.

(Joint work with Oleksiy Klurman, Pär Kurlberg and Joni Teräväinen)

Colloquium on 17 February 2023 at 16:00

Speaker: Clinton Conley (Carnegie Mellon)
Title: Borel reducibility

Abstract: The idea of comparing the complexity of two equivalence relations on the reals by searching for a _Borel reduction_ from one to the other has transformed the last three decades of descriptive set theory. In addition to making precise the intuitive notion that some classification problems are "harder" than others, it has inspired fruitful connections with other areas of math such as dynamical systems and combinatorics.

We give an introduction to this theory, working through some specific examples and sampling some recent developments. Particular attention will be paid to how group- and graph-theoretic notions manifest in the Borel reducibility hierarchy.

Combinatorics on 17 February 2023 at 14:00

Speaker: Mehtaab Sawhney (MIT)
Title: The existence of subspace designs

Abstract: We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility conditions. This settles an open problem from the 1970s. Moreover, we also obtain an approximate formula for the number of such designs. Joint w. Peter Keevash and Ashwin Sah.

Applied Mathematics on 17 February 2023 at 12:00

Speaker: Bindi Brook (Nottingham)
Title: Airway remodelling in asthma - the chicken or the egg?

Abstract: Inflammation, airway hyper-responsiveness (which causes constriction of the airways at lower trigger levels than in normal subject) and airway remodelling (long term structural changes of the airway wall) are key features of asthma. While this is well-established, it is not clear how they are linked or whether they are causes or symptoms of the disease. In this talk I will give an overview of the multiscale biomechanical models we have developed to understand how smooth muscle contraction at the cell level translates to tissue level airway constriction during an asthma attack. Then I will describe a long-timescale theoretical model, developed in parallel with an experimental study, that accounts for mechanochemical drivers of airway remodelling with some illustrative results. And finally I will describe some work in progress: how the combination of both experimental data and the mechanistic model might be used to understand maintenance of homoeostasis in healthy airways and therefore what perturbations might drive the airway into a diseased state.

Probability Theory on 15 February 2023 at 16:00

Speaker: Ellen Powell (University of Durham)
Title: Brownian excursions, conformal loop ensembles and critical Liouville quantum gravity

Abstract: It was recently shown by Aidekon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.

Postgraduate on 15 February 2023 at 12:00

Speaker: Eva Zaat (University of Warwick)
Title: Mathematical modelling of metal forming

Abstract: Metal sheets are everywhere around us; they are used in the architecture of buildings, the manufacturing of transportation and decorative art. Metal sheets are versatile due to the many sizes and shapes they can be deformed in. In this talk we will look into the maths behind metal forming.
We will zoom in on the physical mechanics of elastic bending of sheets. By using a toy problem we will be able to recover the governing equations for a simplified setting. These can be expanded to a system of fourth order non-linear PDEs that describe the deformation of metals in a continuum mechanics framework. We will finish by looking at more complicated problems and the mathematical challenges they pose.

Ergodic Theory and Dynamical Systems on 14 February 2023 at 14:00

Speaker: Caroline Series (University of Warwick)
Title: Convergence of spherical averages for Fuchsian groups

Abstract: Suppose given a measure preserving action of a Fuchsian group on a probability space X, together with a real valued function f on X. We prove pointwise convergence of spherical averages, more precisely, averages of f(gx) over all words of length 2n in a fixed set of generators.
We will briefly review previous results which involve either Cesàro averages or are restricted to free groups. The current proof is based on a new variant of the Bowen--Series symbolic coding for Fuchsian groups that simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by Bufetov in the free group case may be applied.
This is joint work with Sasha Bufetov and Alexey Klimenko, to appear in Commentarii Math Helvetica.

Algebra on 13 February 2023 at 17:00

Speaker: Dylan Johnston (University of Warwick)
Title: Homotopy type of SL2 quotients of simple complex Lie groups

Abstract: We say an element X in a Lie algebra g is nilpotent if ad(X) is a nilpotent operator. It is known that G_{ad}-orbits of nilpotent elements of a complex semisimple Lie algebra g are in 1-1 correspondence with Lie algebra homomorphisms \phi: sl2 -> g, which are in turn in 1-1 correspondence with Lie group homomorphisms SL2 -> G.
Thus, we may denote the homogeneous space obtained by quotienting G by the image of a Lie group homomorphism SL2 -> G by X_v, where v is a nilpotent element in the corresponding G_{ad}-orbit.
In this talk we introduce some algebraic and topological tools that one can use to attempt to classify the homogeneous spaces, X_v, up to homotopy equivalence.

Number Theory on 13 February 2023 at 15:00

Speaker: Nikoleta Kalaydzhieva (UCL)
Title: Properties of the multiple solutions to the polynomial Pell equation

Abstract: In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell's equation for $D$. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions $(u_n,v_n)$ ($n\in\mathbb{Z}$). Our object of interest is the polynomial version of Pell’s equation, where the integers are replaced by polynomials with complex coefficients. We investigate the factors of $v_n(t)$. In particular, we show that over the complex polynomials, there are only finitely many values of n for which $v_n(t)$ has a repeated root. Restricting our analysis to $\mathbb{Q}[t]$, we give an upper bound on the number of 'new' factors of $v_n(t)$ of degree at most $N$. Furthermore, we show that all 'new' linear rational factors of $v_n(t)$ can be found when $n \leq 3$, and all 'new' quadratic rational factors when $n \leq 6$.

Combinatorics on 10 February 2023 at 14:00

Speaker: Nóra Frankl (Open University)
Title: Helly numbers of exponential lattices

Abstract: The Helly number of a set in the plane is the smallest N such that the following is true. If any N members of a finite family of convex sets contains a point of S, then there is a point of S which is contained in all members of the family. An exponential lattice with base x consists of points whose coordinates are positive integer powers of x. We prove lower and upper bounds on Helly numbers of exponential lattices in terms of x, and we determine their values exactly in some cases. We also consider asymmetric exponential lattices, and characterise those that have finite Helly numbers. Joint work with Gergely Ambrus, Martin Balko, Attila Jung and Márton Naszódi.

Applied Mathematics on 10 February 2023 at 12:00

Speaker: Yunan Yang (ETH)
Title: Benefits of Weighted Training in Machine Learning and PDE-based Inverse Problems

Abstract: Many models in machine learning and PDE-based inverse problems exhibit intrinsic spectral properties, which have been used to explain the generalization capacity and the ill-posedness of such problems. In this talk, we discuss weighted training for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge of the object to be learned and a strategy to weight the contribution of training data in the loss function. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model in both machine learning and PDE-based inverse problems.

Junior Analysis and Probability Seminar on 09 February 2023 at 13:00

Speaker: William O'Regan (University of Warwick)
Title: A selected survey of projection theorems

Abstract: TBA

Probability Seminar on 08 February 2023 at 16:00

Speaker: Nikolay Barashkov (University of Helsinki)
Title: Invariant measure for the Anderson wave equation.

Abstract: The Anderson Hamiltonian is an operator whose potential is given by white noise. The singular nature of the potential requires renormalization, but nevertheless it can be made sense of as a self adjoint operator. In this talk we will study the wave equation associated to the Anderson Hamiltonian. Since Bourgain's work there has been a program of constructing invariant measures for dispersive equations. In this talk we will carry this out for the Anderson wave equation. A key part of the proof is the coupling of an "Anderson GFF" with the "Standard" Gaussian free field.

Algebraic Geometry on 08 February 2023 at 15:00

Speaker: Inder Kaur (Loughborough)
Title: Specialization of dominant maps

Abstract: In the last few years many results on the specialization of rationalilty have been shown. In this talk I will discuss the concept of irrationality and give a short overview of related results. I will then discuss the specialization of dominant maps in smooth families.

Algebraic Topology on 07 February 2023 at 16:00

Speaker: Luca Pol (Regensburg)
Title: Quillen stratification in equivariant homotopy theory

Abstract: Quillen’s celebrated stratification theorem provides a geometric description of the Zariski spectrum of the cohomology ring of any finite group with coefficients in a field in terms of information coming from its elementary abelian p-subgroups. The goal of this talk is to discuss an extension of Quillen's result to the world of equivariant tensor-triangular geometry. For the category of equivariant modules over a commutative equivariant ring spectrum we obtain a stratification result in the terms of the geometric fixed points equipped with their Weyl-group actions for all subgroups, and hence a classification of localizing tensor ideals. Finally, I will apply these methods to several examples of interests such as Borel-equivariant Morava E-theory and equivariant topological K-theory. This is joint work with Tobias Barthel, Natalia Castellana, Drew Heard and Niko Naumann.

Algebra on 06 February 2023 at 17:00

Speaker: Alastair Litterick (University of Essex)
Title: Representation varieties and rigidity in finite simple groups

Abstract: Building from the now-classical theorem that every non-abelian finite simple group is 2-generated, one can ask much more delicate questions, for instance on the abundance of generating pairs, or on the existence of generating pairs with particular orders, or from particular conjugacy classes.

For groups of Lie type, these questions can be studied using the representation variety Hom(F,G) where F is finitely generated and G is a reductive algebraic group. In work with Ben Martin (Aberdeen), we use the conjugation action of G on Hom(F,G) to interpret generators for groups of Lie type as certain Zariski-closed orbits. This allows us to prove and generalise a 2010 conjecture of C. Marion, and motivates this as an avenue in the wider study of generating sets in groups of Lie type.

Number Theory on 06 February 2023 at 15:00

Speaker: Amitay Kamber (Cambridge)
Title: Optimal Lifting for SL_n(Z)

Abstract: Let q be a natural number. The strong approximation theorem for $SL_n(Z)$ says that the modulu $q$ map $SL_n(Z) \to SL_n(Z/qZ)$ is onto.

This leads to the following research problem: Given a parameter T, look at the (finite) set of matrices $B_T:={A\in SL_n(Z) : ||A|| \le T}$, where $||.||$ is some matrix norm. We are interested in understanding the image of $B_T$ in $SL_n(Z/qZ)$, for T a function of q. Such studies were initiated (in a more general context) by Duke–Rudnick–Sarnak, and further developed by many others, notably Gorodnik–Nevo.

We will focus on the problem of covering $SL_n(Z/qZ)$ with the image of $B_T$, and explain the connection of the problem to the Generalized Ramanujan Conjecture in automorphic forms.

Based on a joint work with Subhajit Jana.

Combinatorics on 03 February 2023 at 14:00

Speaker: Shumin Sun (Warwick)
Title: Factors in quasi-random hypergraphs

Abstract: Given two k-graphs (k-uniform hypergraphs) F and H, an F-factor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi studied the F-factor problem in quasi-random k-graphs with minimum degree. They posed the problem of characterizing the k-graphs F such that every sufficiently large quasi-random k-graph with constant edge density and minimum degree contains an F-factor, and in particular, they showed that all linear k-graphs satisfy this property. We prove a general theorem on F-factors which reduces the F-factor problem of Lenz and Mubayi to a natural sub-problem F-cover problem. By using this result, we answer the question of Lenz and Mubayi for those F which are k-partite k-graphs, and for all 3-graphs.

This is joint work with Laihao Ding, Jie Han, Guanghui Wang, and Wenling Zhou.

Algebraic Topology on 31 January 2023 at 16:00

Speaker: Neil Strickland (Sheffield)
Title: Questions around chromatic splitting

Abstract: The Chromatic Splitting Conjecture of Hopkins states that if we take
the sphere spectrum, localise with respect to the Morava K-theory
K(n), then localise again with respect to E(n-1), then the result
splits as a coproduct of 2^n pieces, each of which is a sphere
localised with respect to E(m) for some m < n. This is known to be
false (by work of Beaudry) when n=p=2, but remains open (and
mysterious) in general. We will explain how the conjecture, when
combined with known phenomena such as chromatic fracture squares,
predicts some calculations with a very intricate
combinatorial/algebraic structure. These calculations appear to be
self-consistent, which could easily have failed to be the case; this
suggests that the conjecture may be true, or closely related to the
truth.

Algebra on 30 January 2023 at 17:00

Speaker: Miriam Norris (University of Manchester)
Title: Some composition multiplicities for tensor products of irreducible representations of GL(n)

Abstract: Understanding the composition factors of tensor products is an important question in representation theory. In characteristic 0 the classical Littlewood-Richardson coefficients describe the composition factors of both the tensor products of simple CGLn(C)-modules and the restriction of simple CGLn(C)-modules to some Levi subgroup.
Now let F denote an algebraically closed field of characteristic p > 0. In comparison very little is known about composition factors of tensor products of simple FGLn(F)-modules but it is thought that there may still be a relationship with the restriction of simple FGLn(F)-modules to some Levi subgroup. In this talk explore and explicit relationship of this kind for tensor products of simple FGLn(F)-modules with the wedge square of the dual natural module and see how this might be used to find composition factors.

Number Theory on 30 January 2023 at 15:00

Speaker: Alex Walker (UCL)
Title: Sums of Hecke Eigenvalues in a Quadratic Sequence

Abstract: Many arithmetic functions which are well-understood on average over sets of positive density remain mysterious when considered over sparser sets. For example, it is not known if there are infinitely many primes of the form n^2 + 1. The behavior of the divisor function on quadratic sequences was first studied by Hooley and refined by Bykovskii. More recently, Blomer has asked a similar question for the Hecke eigenvalues of a holomorphic cusp form. In this talk, we show how to strengthen Blomer's error estimate through the use of shifted convolution sums and the spectral theory of (half-integral weight) automorphic forms.

Colloquium on 27 January 2023 at 16:00

Speaker: Gianluca Crippa (Basel)
Title: Anomalous dissipation in fluid dynamics

Abstract: Kolmogorov's K41 theory of fully developed turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids: although smooth solutions of the Euler equations conserve the energy, in a turbulent regime information is transferred to small scales and dissipation can happen even without the effect of viscosity, and it is rather due to the limited regularity of the solutions. In rigorous mathematical terms, however, very little is known. In a recent work in collaboration with M. Colombo and M. Sorella we consider the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. In my talk, I will present the general context and illustrate the main ideas behind our construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. I will also describe how the same techniques provide an example of lack of selection for passive-scalar advection under vanishing diffusivity, and an example of anomalous dissipation for the forced Euler equations in the supercritical Onsager regularity regime (this last result has been obtained in collaboration with E. Brue', M. Colombo, C. De Lellis, and M. Sorella).

Combinatorics on 27 January 2023 at 14:00

Speaker: Freddie Illingworth (University of Oxford)
Title: Reconstructing a point set from a random subset of its pairwise distances

Abstract: Let $V$ be a set of $n$ points on the real line whose positions are not known. Suppose the distances between pairs of points are revealed one-by-one in a uniformly random order. When is it possible to reconstruct a linear portion/almost all/all of $V$ up to isometry? In this talk I will discuss the thresholds for this problem as well as giving a precise hitting time result for complete reconstruction of $V$. This is strongly related to the notion of global rigidity for graphs.

This is based on joint work with António Girão, Lukas Michel, Emil Powierski, and Alex Scott.

Applied Mathematics on 27 January 2023 at 12:00

Speaker: Graham Benham (Oxford)
Title: From Olympic rowing to gunwale bobbing: Wave drag and wave thrust phenomena

Abstract: In this talk I will discuss different wave phenomena associated with motion on the surface of water. I’ll start with some previous work on waves in Olympic rowing and how to reduce drag, either by tuning the shape of the boat, or by exploiting hysteresis patterns in regions of shallow water. Then I’ll talk about current research within the theme of wave-driven propulsion, i.e. how thrust can be generated from surfing the gradients of a self-generated wave-field. This phenomenon occurs across a wide range of scales, from tiny walking droplets in a vibrating bath, to jumping up and down on the sides of a canoe to drive it forwards, also known as gunwale bobbing.

Geometry and Topology on 26 January 2023 at 14:00

Speaker: Davide Spriano (University of Oxford)
Title: Combinatorial criteria for hyperbolicity

Abstract: Perhaps one of the most fascinating properties of hyperbolic groups is that they admit equivalent definitions coming from different areas of mathematics. In this talk, we will survey some interesting definitions, and discuss a new one that, perhaps surprisingly, was previously unknown, namely that fact that hyperbolicity can be detected by the language of quasi-geodesics in the Cayley graph. As an application, we will discuss some progress towards a conjecture of Shapiro concerning groups with uniquely geodesic Cayley graphs.

Junior Analysis and Probability Seminar on 26 January 2023 at 13:00

Speaker: Eduardo Tablate (ICMAT)
Title: Schur multipliers in Schatten von Neumann classes and noncommutative Fourier multipliers

Abstract: TBA

Probability Seminar on 25 January 2023 at 16:00

Speaker: Vittoria Silvestri (Università di Roma La Sapienza)
Title: Explosive growth for a constrained Hastings–Levitov aggregation model

Abstract: The Hastings--Levitov (HL) growth models describe the formation of random aggregates in the complex plane via conformal maps. In this talk I will discuss a version of the HL models on the upper half plane, in which the growth is restricted to the cluster boundary. We will see that, although one might expect a shape theorem, this constrained model exhibits explosive behaviour, in that the cluster accumulates infinite diameter as soon as it reaches positive capacity. Based on joint work with Nathanael Berestycki.

Algebraic Geometry on 25 January 2023 at 15:00

Speaker: Roberto Gualdi (Regensburg)
Title: How complicated are the solutions of a system of polynomial equations?

Abstract: A beautiful result due to Bernstein and Kushnirenko allows to predict the number of solutions of a system of Laurent polynomial equations from the combinatorial properties of the defining Laurent polynomials.

In a joint work with Martín Sombra (ICREA and Universitat de Barcelona), we give intuitions for an arithmetic version of such a theorem. In particular, in the easy case of the planar curve x + y + 1 = 0, we show how to guess the arithmetic complexity of the intersection point between this line and its translates by torsion points.

The talk will involve a bit of height theory, special values of the Riemann zeta function and will not forget to pay homage to a piece of British literature.

Ergodic Theory and Dynamical Systems on 24 January 2023 at 14:00

Speaker: Xiong Jin (Manchester University)
Title: An extension of Hochman and Shmerkin’s projection theorem

Abstract: In this talk I will present an extension of Hochman and Shmerkin’s projection theorem on the product of integer-multiplication invariant measures on the unit circle. In the symbolic setting we extend these measures to left-shift invariant measures mapped through one-dimensional iterated function systems without any separation conditions. Consequently, we prove that Bernoulli convolutions with log-rationally independent parameters are dissonate, i.e., their convolution has the maximal possible dimension. If time allows, I will also mention the extension of H.-S. theorem from invariant measures to a class of random measures called Mandelbrot cascades. This leads to an extension of Furstenberg’s sumset conjecture (now a theorem by Hochman and Shmerkin) to some more general random fractal sets.

Algebra on 23 January 2023 at 17:00

Speaker: Michael Bate (University of York)
Title: Simple Modules for Algebraic Groups

Abstract: I will report on current work with David Stewart. In this work we have: a) given a classification of simple modules for algebraic groups over arbitrary fields, which extends well-known high weight classifications when the field is algebraically closed and/or the group is reductive; b) begun to explore the structure of these simple modules, in particular how they look after extensions of the ground field. The key to progress is to understand so-called "pseudo-reductive groups". I will spend most of the talk on an extended example of such a group and its representations, which can be constructed pretty concretely from the simple algebraic group SL_2.

Number Theory on 23 January 2023 at 15:00

Speaker: Elisa Lorenzo García (Neuchâtel)
Title: Lower bound on the maximal number of rational points on curves over finite fields

Abstract: For a long time people have being interested in finding and constructing curves with many points. For genus 1 and genus 2 curves, we know how to construct curves over any finite field of defect less than 1 or 3 (respectively), i.e. with a number of points at distance at most 1 or 3 to the upper bound given by the Hasse–Weil–Serre bound. The case of genus 3 is still open after more than 40 years of research. In this talk I will take a different approach based on the random matrix theory of Katz–Sarnak to prove the existence, for all \epsilon > 0, of curves of genus g over Fq with more than 1 + q + (2g −\epsilon )\sqrt{q} points for q big enough. I will also discuss some explicit constructions as well as some consequences to the Serre obstruction problem (an asymmetric behaviour of the distribution of the trace of the Frobenius for curves of genus 3).

This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler.

Colloquium on 20 January 2023 at 16:00

Speaker: Daniel Meyer (Liverpool)
Title: Fractal spheres, visual metrics, and rational maps

Abstract: The aim of this talk is to show parallels between different areas of mathematics, namely between complex dynamics, Kleinian groups, and random geometry. A common theme are certain fractal spheres that arise. These are the Brownian map (in the random setting) and spheres equipped with so-called visual metrics in the dynamical setting.
A relevant class of maps in this context are quasisymmetric maps. These map ratios of distances in a controlled way and generalize conformal maps.
A Thurston map is a topological analog of a rational map (i.e., a holomorphic self-map of the Riemann sphere). Thurston gave a criterion when such a map ``is'' rational. Given such a map f that is expanding, we can equip the sphere with a "visual metric". With respect to this metric, the sphere is a quasisphere if and only if f "is" rational.
This is joint work with Mario Bonk (UCLA).

Combinatorics on 20 January 2023 at 14:00

Speaker: Amedeo Sgueglia (UCL)
Title: A general approach to rainbow versions of Dirac-type theorems

Abstract: Given a collection of m hypergraphs on the same vertex set V, a rainbow copy of an m-edge graph F is a copy of F on V obtained by selecting exactly one edge from each hypergraph of the collection. How large does the minimum degree of each graph in the collection need to be so that it necessarily contains a rainbow copy of F? Each hypergraph in the collection could be the same hypergraph, hence the minimum degree of each of them needs to be large enough to ensure that it individually contains F. In this talk, we discuss a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel rainbow variants of several classical Dirac-type results for (powers of) Hamilton cycles.

This is joint work with Pranshu Gupta, Fabian Hamann, Alp Müyesser, and Olaf Parczyk.

Applied Mathematics on 20 January 2023 at 12:00

Speaker: Bryn Davies (Imperial College)
Title: Are quasicrystals the future of metamaterial waveguides?

Abstract: Characterising the spectra of quasiperiodic patterns is a challenging problem that has attracted the attention of mathematicians for several decades. Metamaterial waveguides achieve spectacular wave control feats through carefully designed geometric patterns. In most applications, these patterns are typically periodic, meaning their spectra can be characterised concisely using Floquet-Bloch techniques. However, modern techniques for describing quasiperiodic patterns are now sufficiently developed that we can use them when designing metamaterial waveguides for specific applications, thereby greatly enlarging the potential design space. This talk will summarise some recent breakthroughs in the characterisation of spectral gaps in quasiperiodic metamaterials and identify opportunities where this theory can be used in applications. In particular, we will look at applications of quasicrystals to graded energy harvesting devices and symmetry-induced waveguides.

Junior Analysis and Probability Seminar on 19 January 2023 at 13:00

Speaker: Iain Souttar (Heriot-Watt University)
Title: Uniform in time approximations: Averaging

Abstract: TBA

Algebraic Geometry on 18 January 2023 at 15:00

Speaker: Qaasim Shafi (Birmingham)
Title: Divisors on Logarithmic Mapping Spaces

Abstract: The stable maps compactification of the space of rational, degree d curves in P^r is used to define Gromov-Witten invariants and is helpful for solving enumerative problems concerning rational curves in projective space. Its geometry is well studied. In particular, its Picard/Class group (over Q) was determined by Pandharipande.

The space of rational, degree d curves in P^r with fixed tangencies to a hyperplane H has a compactification by stable logarithmic maps and is used to define relative (or logarithmic) Gromov-Witten invariants. In joint work with Patrick Kennedy-Hunt, Navid Nabijou and Wanlong Zheng we determine its Class group and Picard group (over Q). I will highlight some of the differences which arise in the logarithmic case and explain how this is part of a broader programme aimed at understanding the geometry of logarithmic mapping spaces.

Postgraduate on 18 January 2023 at 12:00

Speaker: Peize Liu (University of Warwick)
Title: Introduction to deformation quantisation and formality

Abstract: In 1997, Kontsevich solved the problem of deformation quantisation on Poisson manifolds, which contributed to his winning of the 1998 Fields Medal.

This talk is an introduction to deformation quantisation. This is an approach of going from classical mechanics to quantum mechanics through deformation of the algebra of smooth functions on the phase space. I will explore the original idea from physics and go through some historical developments. Then I will give a crash course on deformation theory based on differential graded Lie algebras and L∞-algebras, and show its connection with deformation quantisation via Kontsevich’s formality theorem.

Algebraic Topology on 17 January 2023 at 16:00

Speaker: Foling Zou (University of Michigan)
Title: Nonabelian Poincare duality theorem in equivariant factorization homology

Abstract: The factorization homology are invariants of n-dimensional manifolds with some fixed tangential structures that take coefficients in suitable En-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group G by monadic bar construction following Kupers--Miller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by Dotto--Moi--Patchkoria--Reeh.

Number Theory on 16 January 2023 at 15:00

Speaker: Julia Stadlmann (Oxford)
Title: The mean square gap between primes

Abstract: Conditional on the Riemann hypothesis, Selberg showed in 1943 that the average size of the squares of differences between consecutive primes less than x is O(log(x)^4). Unconditional results still fall far short of this conjectured bound: Peck gave a bound of O(x^{0.25+epsilon}) in 1996 and to date this is the best known bound obtained using only methods from classical analytic number theory.

In this talk we discuss how sieve theory (in the form of Harman's sieve) can be combined with classical methods to improve bounds on the number of short intervals which contain no primes, thus improving the unconditional bound on the mean square gap between primes to O(x^{0.23+epsilon}).

Colloquium on 13 January 2023 at 16:00

Speaker: Tom Hudson (Warwick)
Title: Recent developments in the modelling and theory of crystalline defects

Abstract: Crystalline solids are found all around us, and are made up of atoms arranged in a translation-invariant structure. Although this is a significant part of the story for such materials, crystals also contain many defects which break this symmetry, and it is these defects which turn out to be crucial in determining many of the physical properties of a crystalline material. I will begin by motivating the study of such defects, and provide an overview of the range of physical theories used to model and simulate their behaviour. I will then discuss some of the important recent developments in this area (many of which have a Warwick connection), including both novel machine-learning-based approaches and rigorous mathematical developments.

Combinatorics on 13 January 2023 at 14:00

Speaker: Christian Ikenmeyer (University of Warwick)
Title: Characters of the symmetric group and combinatorial interpretations

Abstract: The character of the symmetric group evaluated at the identity is just the number of standard Young tableaux. We study what happens if we evaluate at other elements. Here the character can attain negative values. Our main result is that the square of the character of the symmetric group has no (unsigned) combinatorial description, unless the polynomial hierarchy collapses to the second level. This also works for the absolute value instead of the square.

Applied Mathematics on 13 January 2023 at 12:00

Speaker: Fabian Spill (Birmingham)
Title: Mechanics, Geometry and Topology of Health and Disease

Abstract: Experimental biologists traditionally study biological functions as well as diseases mostly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. However, many diseases also have a mechanical component which is critical to their deadliness. Notably, cancer kills mostly through metastasis, where the cancer cells acquire the capability to change their physical attachments and migrate. Such mechanical alterations also change geometrical features, such as the cell shape, or topological features, such as the organisation of vascular networks and cellular neighbourhoods within a tissue.

While some of these mechanical, geometrical or topological features in biology are long known, the traditional perspective is to consider them as emergent from molecular features. However, mechanical, geometrical and topological features can also affect the molecular state of a cell. Therefore, the most complete view of many biological systems is to consider them as a complex mechano-chemical systems. Diseases such as cancer are then interpreted as perturbations to this system that cannot be solely explained by considering one feature in isolation (such as a single mutation that ‘causes’ cancer).

I will discuss several examples of systems where this mechanical/geometrical/topological coupling to molecular features plays a crucial role: cells that change their shape, blood vessel cells that open gaps to let cancer cells pass during metastasis, and mitochondria that change their organisation in diabetes.

Algebraic Geometry on 11 January 2023 at 15:00

Speaker: Jenia Tevelev (University of Massachusetts, Amherst)
Title: Semi-orthogonal decompositions of moduli spaces

Abstract: Let C be a smooth projective curve of genus g at least 2 and let N be the moduli space of stable rank 2 vector bundles on C with fixed odd determinant. It is a smooth Fano variety of dimension 3g-3, Picard number 1 and index 2. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i = 0,...,g-2 and one block for the (g-1)-st symmetric power. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. Joint work with Sebastian Torres.

Postgraduate on 11 January 2023 at 12:00

Speaker: Hollis Williams (University of Warwick)
Title: Fourier analysis for rarefied gas flows

Abstract: Fourier analysis is widely used in applied mathematics, engineering and physics. In this talk, we explain how it can be used to derive some new exact solutions for non-equilibrium rarefied gas flows. These flows fall into a regime which is inaccessible both to the Boltzmann and Navier-Stokes equations, so a different set of equations must be used known as the Grad equations.

Algebraic Topology on 10 January 2023 at 16:00

Speaker: Ismael Sierra (Cambridge)
Title: Homological stability of diffeomorphism groups using Ek algebras

Abstract: I will state some recent results about homological stability of diffeomorphism groups of manifolds and give an outline of their proof. In particular, I will talk about the connection to Ek algebras, and about certain complexes, called "splitting complexes", whose high-connectivities are essential to the proof. Finally I will sketch the proof of the high-connectivity of the splitting complexes, which is the most substantial part of the whole argument.

Ergodic Theory and Dynamical Systems on 10 January 2023 at 14:00

Speaker: Cagri Sert (University of Warwick)
Title: Stationary measures for SL(2,R)-actions on homogeneous bundles over flag varieties

Abstract: Let X_{k,d} denote the space of rank-k lattices in R^d. Topological and statistical properties of the dynamics of discrete subgroups of G = SL(d,R) on X_{d,d} were described in the seminal works of Benoist--Quint. A key step/result in this study is the classification of stationary measures on X_{d,d}. Later, Sargent--Shapira initiated the study of dynamics on the spaces X_{k,d}. When k < d, the space X_{k,d} is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure \mu Zariski-dense in a copy of SL(2,R) in G, we give a classifi cation of stationary measures on X_{k,d} and prove corresponding equidistribution results. In contrast to the results of Benoist--Quint, the type of stationary measures that \mu admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will review the preceding works (Benoist--Quint, Eskin--Lindenstrauss, Sargent--Shapira) and discuss main cases and ideas. Joint work with Alexander Gorodnik and Jialun Li.

Number Theory on 09 January 2023 at 15:00

Speaker: Wissam Ghantous (Oxford)
Title: A symmetric triple product p-adic L-function

Abstract: In 2014, Darmon and Rotger define the Garrett–Rankin triple product p-adic L-function and relate it to the image of certain diagonal cycles under the p-adic Abel–Jacobi map. We introduce a new variant of this p-adic L-function and show that it satisfies symmetry relations, when permuting the three families of modular forms. We also provide computational evidence confirming that it is indeed cyclic when the families of modular forms are evaluated at even weights, and provide counter-examples in the case of odd weights. To do so, we extend Lauder's algorithm (for computing ordinary projections of nearly overconvergent modular forms) to work with nearly overconvergent modular forms and compute projections over spaces of non-zero slope.

Colloquium on 09 December 2022 at 16:00

Speaker: John Baez (UC Riverside)
Title: Category Theory in Epidemiology

Abstract: Category theory provides a general framework for building models of dynamical systems. We explain this framework and illustrate it with the example of "stock and flow diagrams". These diagrams are widely used for simulations in epidemiology. Although tools already exist for drawing these diagrams and solving the systems of differential equations they describe, we have created a new software package called StockFlow which uses ideas from category theory to overcome some limitations of existing software. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

Combinatorics on 09 December 2022 at 14:00

Speaker: Zoltán Vidnyánszky (Eötvös Loránd University)
Title: Borel combinatorics, the LOCAL model and complexity

Abstract: In the first part of the talk, I will give an overview of the field of Borel combinatorics and its recently uncovered connections to the LOCAL model of distributed computing. Then, I will discuss complexity related aspects of the field. Namely, I will consider the question of how hard it is to decide the existence of Borel homomorphisms from a Borel structure to a given finite structure.

Geometry and Topology on 08 December 2022 at 14:00

Speaker: Ric Wade (University of Oxford)
Title: Aut-invariant quasimorphisms on groups

Abstract: For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. Joint work with Francesco Fournier-Facio.

Probability Seminar on 07 December 2022 at 16:00

Speaker: Adrián Gonzáles Casanova (UNAM)
Title: Sampling Duality

Abstract: Sampling Duality is stochastic duality using a duality function S(n,x) of the form ¨what is the probability that all the members of a sample of size n are of type -, given that the number (or frequency) of type - individuals is x¨. Implicitly this technique can be traced back to the work of Pascal. Explicitly it is studied in a paper of Martin Möhle in 1999. We will discuss several examples in which this technique is useful, including Haldane's formula and the long standing open question of the rate of the Muller Ratchet.

Postgraduate on 07 December 2022 at 12:00

Speaker: Elvira Lupoian (University of Warwick)
Title: Jacobians of curves: A brief introduction

Abstract: To any algebraic curve C of genus g, we can associate its Jacobian, a g-dimensional abelian variety which is functorially associated to the curve. In this talk, I will define Jacobians, assuming no previous knowledge in the subject and explore some of their properties. If time permits, I will touch on one of the ways in which rational points on a Jacobian can be used to find the set of rational points on the corresponding curve.

Algebraic Topology on 06 December 2022 at 16:00

Speaker: Lucy Yang (Harvard)
Title: A real Hochschild--Kostant--Rosenberg theorem

Abstract: Grothendieck--Witt and real K-theory are enhancements of K-theory in the presence of duality data. Similarly to ordinary K-theory, real K-theory admits homological approximations, known as real trace theories. In this talk, I will identify a filtration on real Hochschild homology and compute the associated graded in terms of an analogue of de Rham forms. We will see how C₂ genuine equivariant algebra is the natural setting for these theories, provide equivariant enhancements of the cotangent and de Rham complexes, and sketch the proof of the main theorem. This work is both inspired by and builds on that of Raksit.

Partial Differential Equations and their Applications on 06 December 2022 at 12:00

Speaker: Antonio Esposito (University of Oxford)
Title: TBA

Abstract: TBA

Algebra on 05 December 2022 at 17:00

Speaker: Ana Retegan (University of Birmingham)
Title: TBC

Abstract: TBC

Number Theory on 05 December 2022 at 15:00

Speaker: Istvan Kolossvary (St Andrews)
Title: Distance between natural numbers based on their prime signature

Abstract: One can define different metrics between natural numbers based on their unique prime signature. Fixing such a metric, we are interested in the asymptotic growth rate of the arithmetic function L(N) which tabulates the cumulative sum of distances between consecutive natural numbers up to N. In particular, choosing the maximum norm, we will show that the limit of L(N)/N exists and is equal to the expected value of a certain random variable. We also demonstrate that prime gaps exhibit a richer structure than on the traditional number line and pose a number of problems. Joint work with Istvan B. Kolossvary.

Colloquium on 02 December 2022 at 16:00

Speaker: Tara Brendle (Glasgow)
Title: Twists and trivializations: encoding symmetries of manifolds

Abstract: The classification of 2-manifolds in the first half of the 20th century was a landmark achievement in mathematics, as was the more recent (and more complicated) classification of 3-manifolds completed by Perelman. The story does not end with classification, however: there is a rich theory of symmetries of manifolds, encoded in their mapping class groups. In this talk we will explore some aspects of mapping class groups in dimensions 2 and 3, with a focus on illustrative examples.

Combinatorics on 02 December 2022 at 14:00

Speaker: Bernd Schulze (University of Lancaster)
Title: Geometric Rigidity Theory and Applications

Abstract: In the last two decades or so the subject has become particularly active, drawing on diverse areas of mathematics, and engaging with a growing range of modern applications, such as Engineering, Robotics, Computer-Aided-Design, Molecular Dynamics, and Materials Science.

In the first part of the talk, I will give an introduction to Geometric Rigidity Theory, concentrating on some key combinatorial results and problems for bar-joint frameworks, but also describing how these have been extended to some other types of frameworks.

Since many real-world structures are symmetric, a major recent research direction in the field is to study the impact of symmetry on the rigidity and flexibility of bar-joint frameworks. I will show how group representation theory can be used to reveal `hidden' infinitesimal motions and states of self-stress in symmetric frameworks that cannot be detected with the standard non-symmetric counts. Finally, I will show how these symmetry-based methods can be used as a design tool for gridshell structures. This is recent joint work with William Baker, Arek Mazurek and Cameron Millar.

Applied Mathematics on 02 December 2022 at 12:00

Speaker: Katherine Kamal (Cambridge)
Title: The microhydrodynamics of ultra-thin nanoparticles: modelling to predict the "unseen"

Abstract: Graphene nanoparticles are ubiquitous, used in everything from the design of more robust extreme weather-resistance spacecraft to flexible-electronics tracks. Made from just a few atomic layers, the instantaneous dynamics of these plate-like particles in flowing liquids are, experimentally, practically inaccessible. We study theoretically and computationally the microhydrodynamics of dilute suspensions of graphene in a simple viscous shear flow field. In the infinite Péclet number limit, a rigid platelet with the interfacial hydrodynamic slip properties of graphene does not follow the periodic rotations predicted for classical colloidal particles but aligns itself at a slight inclination angle with respect to the flow. This unexpected result is due to the hydrodynamic slip reducing the tangential stress at the graphene-liquid surface. By analysing the Fokker-Plank equation for the orientational distribution function for decreasing Péclet numbers, we explore how hydrodynamic slip affects the particle’s orientation and effective viscosity. We find that hydrodynamic slip can dramatically change the average particle’s orientation and effective viscosity. For example, the effective viscosity of a dilute suspension of graphene platelets is predicted to be smaller than the base fluids under certain flow conditions for typical slip length values.

Probability Seminar on 01 December 2022 at 16:00

Speaker: Sunil Chhita (University of Durham)
Title: Domino Shuffle and Matrix Refactorizations

Abstract: This talk is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).

Geometry and Topology on 01 December 2022 at 14:00

Speaker: Koji Fujiwara (Kyoto University)
Title: The rates of growth in a hyperbolic group

Abstract: I discuss the set of rates of growth of a finitely generated group with respect to all its finite generating sets. In a joint work with Sela, for a hyperbolic group, we showed that the set is well-ordered, and that each number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. I may discuss its generalization to acylindrically hyperbolic groups.

Algebraic Geometry on 30 November 2022 at 15:00

Speaker: Cancelled (n/a)
Title: n/a

Abstract: n/a

Postgraduate on 30 November 2022 at 12:00

Speaker: Nuno Arala Santos (University of Warwick)
Title: Counting Rational Points on Cubic Surfaces

Abstract: A fundamental problem in Diophantine geometry is to understand the asymptotic behaviour of the number of solutions to a Diophantine equation when we impose a boundedness condition on the variables. We will explain some progress in this problem for equations defining cubic surfaces in 3-dimensional space, following Roger Heath-Brown.

Algebraic Topology on 29 November 2022 at 16:00

Speaker: Florian Naef (Trinity College Dublin)
Title: Relative intersection product, Whitehead-torsion and string topology

Abstract: Given a closed oriented manifold one can define an intersection product on the homology. This can be extended to local coefficient, and further made relative to the diagonal. I will explain how such a relative self-intersection product is not homotopy invariant (in contrast to the ordinary intersection product) and how this is picked up by string topology. Eventually, we will identify the error term with the trace of Whitehead torsion. More precisely, we will extract an invariant from a Poincare embedding of the diagonal (in the sense of J. Klein) that is the trace of (a version of) Reidemeister torsion. This is based on joint work with P. Safronov.

Algebra on 28 November 2022 at 17:00

Speaker: Rachel Pengelly (University of Birmingham)
Title: TBC

Abstract: TBC

Number Theory on 28 November 2022 at 15:00

Speaker: Alexandre Maksoud (Paderborn)
Title: The arithmetic of the adjoint of a weight 1 modular form

Abstract: A conjecture of Darmon, Lauder and Rotger expresses p-adic iterated integrals attached to a pair of weight 1 modular forms (f,g) in terms of p-adic logarithms of certain units attached to f and g. This talk reports a work in progress in which we explain, in the case where f=g, how to interpret this conjecture as a variant of the Gross-Stark conjecture for the adjoint of f. This requires studying the specializations of the congruence module attached to a Hida deformation of f.

Colloquium on 25 November 2022 at 16:00

Speaker: Cancelled (-)
Title: -

Abstract: -

Applied Mathematics on 25 November 2022 at 12:00

Speaker: Eric Neiva (Collége de France & CNRS)
Title: Unfitted finite element methods: decoupling the mesh from the geometry

Abstract: The finite element method (FEM) approximates a PDE from a variational formulation of the problem. Its standard formulation requires a mesh fitting to the boundary of the geometry of interest. Yet, for many problems of practical interest, the geometry is so intricate that mesh generation requires frequent and time-consuming manual intervention. Boundary-fitted meshing can be avoided with unfitted or immersed FEMs. The main idea is to embed the geometry in a simple mesh (e.g., a Cartesian grid) and define the discretisation in the cells intersecting the geometry. In this talk, we will describe a novel unfitted FEM that circumvents the classical issue of immersed FEM: ill-conditioning due small cell-to-geometry intersections We will discuss its application to early embryo development in animals.

Junior Analysis and Probability Seminar on 24 November 2022 at 13:00

Speaker: Giacomo del Nin (University of Warwick)
Title: Isoperimetric shapes in Penrose tilings

Abstract: TBA

Probability Seminar on 23 November 2022 at 16:00

Speaker: Pierre-Francois Rodriguez (Imperial College London)
Title: Scaling in low-dimensional long-range percolation models

Abstract: The talk will present recent progress towards understanding the critical behavior of dimensional percolation models exhibiting long-range correlations. The results rigorously exhibit the scaling behavior of various observables of interest and are consistent with scaling theory below the upper-critical dimension (expectedly equal to 6).

Algebraic Geometry on 23 November 2022 at 15:00

Speaker: Jonathan Lai (Imperial)
Title: A Reconciliation of Mutations and Potentials

Abstract: Given a lattice polygon, one can consider the spanning fan to obtain a toric variety. A combinatorial mutation is an operation that takes one polygon to another, which induces a degeneration of one toric variety to the other. One can then attempt to study all toric degenerations of a fixed Fano variety through the study of polygons and their mutations. In another world, a set of algebraic tori can be glued together by birational maps, also called mutations, to form a cluster variety.

In this talk, I will explain a justification coming from mirror symmetry on why these two operations deserve to share the same name (in dimension 2). Given an orbifold del Pezzo surface X, there is a natural cluster variety Y that knows about the polytopes and mutations associated to X. Namely, there is a combinatorial object associated to Y called a scattering diagram, which is a collection of walls inside a vector space. The chambers, which correspond to tori in Y, are precisely the polygons coming from toric degenerations of X. This is based off ongoing joint work with Tim Magee and Ben Wormleighton.

Postgraduate on 23 November 2022 at 12:00

Speaker: Alexandros Groutides (University of Warwick)
Title: Galois representations attached to elliptic curves and the Open Image Theorem

Abstract: A Galois representation is a homomorphism $\rho:Gal(\bar{K}/K)\longrightarrow Aut(V)$ where $V$ is a finite dimensional vector space or a free module of finite rank. These objects are of great importance in number theory due to their connections with elliptic curves, modular forms and $L$-functions. We will introduce the mod-$\ell$, $\ell$-adic and adelic Galois representations attached to a non-CM elliptic curve and discuss the structure of their image. The $\ell$-adic open image does not a priori imply the adelic open image but as we will see, it all boils down to the surjectivity of the more innocent sounding mod-$\ell$ representation.

Algebraic Topology on 22 November 2022 at 16:00

Speaker: Irakli Patchkoria (University of Aberdeen)
Title: Morava K-theory of infinite groups and Euler characteristic

Abstract: Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. Along the way we will give explicit computations for amalgamated products of finite groups, right angled Coxeter groups and certain special linear groups. This is all joint with Wolfgang Lück and Stefan Schwede.

Ergodic Theory and Dynamical Systems on 22 November 2022 at 15:30

Speaker: Tim Austin (UCLA)
Title: Positive sofic entropy without relatively Bernoulli factors.

Abstract: The classical Kolmogorov–Sinai entropy is an invariant of probability-preserving transformations. Much of the resulting theory was successfully extended to actions of discrete amenable groups by Ornstein, Weiss and others.
Lewis Bowen’s more recent notion of sofic entropy extends the Kolmogorov–-Sinai definition to actions of sofic groups, a much larger class introduced by Gromov. A range of natural questions concern how entropy and its consequences differ between the sofic setting the amenable one.
After reviewing a special case of sofic entropy for certain free-product groups, this talk will present a new example of an action of such a group. The example has positive sofic entropy, but has no splitting as a direct product involving a Bernoulli factor. This contrasts with the world of amenable group actions, where many such splittings are guaranteed by the weak Pinsker theorem. The new example is an algebraic action, and its analysis depends on (slight modifications of) results
from the theory of random regular low-density parity-check codes.
This material is part of an ongoing joint project with Lewis Bowen, Brandon Seward and Christopher Shriver.
(This talk will be a continuation of the colloquium from Friday Nov 18th, and will assume some of the notions from that talk.)

Ergodic Theory and Dynamical Systems on 22 November 2022 at 14:00

Speaker: Timothée Bénard (Cambridge)
Title: The local limit theorem for biased random walks on nilpotent groups

Abstract: We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures.

Partial Differential Equations and their Applications on 22 November 2022 at 12:00

Speaker: Annika Bach (Sapienza Università di Roma)
Title: TBA

Abstract: TBA

Algebra on 21 November 2022 at 17:00

Speaker: Diego Martin Duro (University of Warwick)
Title: TBC

Abstract: TBC

Number Theory on 21 November 2022 at 15:00

Speaker: Rachel Greenfeld (IAS)
Title: Aperiodicity of translational tilings

Abstract: Translational tiling is a covering of a space using translated copies of some building blocks, called the "tiles", without any positive measure overlaps. What are the possible ways that a space can be tiled?

A well known conjecture in this area is the periodic tiling conjecture, which asserts that any tile of Euclidean space admits a periodic tiling. In a joint work with Terence Tao, we construct a counterexample to this conjecture. In the talk, I will survey the study of the periodicity of tilings and discuss our recent progress.

Colloquium on 18 November 2022 at 16:00

Speaker: Tim Austin (UCLA)
Title: Some recent developments around entropy in ergodic theory

Abstract: The entropy rate of a stationary sequence of random symbols was introduced by Shannon in his foundational work on information theory in 1948. In the early 1950s, Kolmogorov and Sinai realized that they could turn this quantity into an isomorphism invariant for measure-preserving transformations on a probability space. Almost immediately, they used it to distinguish many examples called "Bernoulli shifts" up to isomorphism. This resolved a famous open question of the time, and ushered in a new era for ergodic theory.

In the decades since, entropy has become one of the central concerns of ergodic theory, having widespread consequences both for the abstract structure of measure-preserving transformations and for their behaviour in applications. In this talk, I will review some of the highlights of the structural story, and then discuss Bowen's more recent notion of `sofic entropy'. This generalizes Kolmogorov--Sinai entropy to measure-preserving actions of many `large' non-amenable groups including free groups. I will end with a recent result illustrating how the theory of sofic entropy has some striking differences from its older counterpart.

This talk will be aimed at a general mathematical audience. Most of it should be accessible given a basic knowledge of measure theory, probability, and a little abstract algebra.

Combinatorics on 18 November 2022 at 14:00

Speaker: Shoham Letzter (UCL)
Title: Separating paths systems of almost linear size

Abstract: A separating path system for a graph G is a collection P of paths in G such that for every two edges e and f, there is a path in P that contains e but not f. We show that every n-vertex graph has a separating path system of size O(n log* n). This improves upon the previous best upper bound of O(n log n), and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O(n) bound should hold.

Applied Mathematics on 18 November 2022 at 12:00

Speaker: Anna Song (Imperial)
Title: Describing tubular shapes and branching membranes with geometry and topology

Abstract: Tubular and membranous shapes are important mathematical structures that arise in many biomedical applications. Morphology is linked to function: their branching patterns, shaped by interactions and remodelled by diseases, inform us on a biological system.

I will present the “curvatubes” model, which unifies a wide continuum of porous shapes within a common geometric framework (https://doi.org/10.1007/s10851-021-01049-9). It generalizes the Helfrich model for biomembranes by considering shapes as optimizers of a curvature functional in which the principal curvatures may play asymmetric roles. The geometric problem is approximated by a novel phase-field formulation that satisfies a Gamma-limsup property, and is readily implementable as a GPU algorithm. The framework is very flexible and shape textures can be aligned, spatialized, or constrained on a domain.

In the remaining time, I will introduce some topological approaches to analyze such structures using persistent homology, and how they may empirically quantify the "texture of shapes". These are tested on proprietary images of bone marrow vasculature remodelled in acute myeloid leukaemia.

Overall, these compact descriptions offer a unified view to branching tubules and membranes, and will potentially lead to applications in bioengineering, imaging, or materials science.

Geometry and Topology on 17 November 2022 at 14:00

Speaker: Bradley Zykoski (Univeristy of Michigan)
Title: A polytopal decomposition of strata of translation surfaces

Abstract: A closed surface can be endowed with a certain locally Euclidean metric structure called a translation surface. Moduli spaces that parametrize such structures are called strata. There is a GL(2,R)-action on strata, and orbit closures of this action are rare gems, the classification of which has been given a huge boost in the past decade by landmark results such as the "Magic Wand" theorem of Eskin-Mirzakhani-Mohammadi and the Cylinder Deformation theorem of Wright. Investigation of the topology of strata is still in its nascency, although recent work of Calderon-Salter and Costantini-Möller-Zachhuber indicate that this field is rapidly blossoming. In this talk, I will discuss a way of decomposing strata into finitely many higher-dimensional polytopes. I will discuss how I have used this decomposition to study the topology of strata, and my ongoing work using this decomposition to study the orbit closures of the GL(2,R)-action.

Junior Analysis and Probability Seminar on 17 November 2022 at 13:00

Speaker: Jakub Takác (University of Warwick)
Title: Norms in finite dimensions and rectifiability in metric spaces

Abstract: TBA

Probability Seminar on 16 November 2022 at 16:00

Speaker: Sam Olesker-Taylor (University of Warwick)
Title: Random Walks on Random Cayley GraphsTBA

Abstract: TBAhttps://teams.microsoft.com/l/meetup-join/19%3ameeting_Mjg1MDU2MDQtY2NlZS00Y2NlLWFhNWUtZWRiMmEwYjI0ZGEz%40thread.v2/0?context=%7b%22Tid%22%3a%2209bacfbd-47ef-4465-9265-3546f2eaf6bc%22%2c%22Oid%22%3a%22325bf9e5-c56a-4d31-9811-aa22fe105e13%22%7d

Join conversation
teams.microsoft.com

We investigate mixing properties of RWs on random Cayley graphs of a finite group G with
k≫ 1 independent, uniformly random generators, with 1 ≪ log k ≪ log |G|.

Aldous and Diaconis (1985) conjectured that the RW on this random graph exhibits cutoff for any group G whenever k ≫ log |G| and further that the cutoff time depends only on k and |G|. It was established for Abelian groups.

We disprove the second part of the conjecture by considering RWs on upper-triangular matrices. We extend this conjecture to 1 ≪ k ≲ log |G|, verifying a version of it for arbitrary Abelian groups under 'almost necessary' conditions on k.

It is all joint work with Jonathan Hermon (now at UBC).

Algebraic Geometry on 16 November 2022 at 15:00

Speaker: Ruijie Yang (Humboldt-Universität zu Berlin)
Title: Zeroes of one forms and homologically trivial deformations

Abstract: In 1926, Hopf proved the Poincaré-Hopf theorem, which implies that if a compact differential manifold admits a nowhere vanishing vector field, then its topological Euler characteristic is zero. Dually, it is natural to ask the same question for one forms. In 1970, Tischler proved that the existence of a nowhere vanishing real closed one form induces a differentiable fiber bundle structure over the circle. In 2013, Kotschick conjectured that for compact Kähler manifolds, admitting a nowhere vanishing real closed one form is actually equivalent to the existence of a nowhere vanishing holomorphic one form. In this talk, I will show that Kotschick’s conjecture can be deduced from a conjecture of Bobadilla-Kollár on homologically trivial deformation. Therefore, Kotschick’s conjecture is true if the first Betti number of X is at least 2dim(X)-2 and the Albanese variety of X is simple. This is joint work with Stefan Schreieder.

Postgraduate on 16 November 2022 at 12:00

Speaker: Andrew Ronan (University of Warwick)
Title: Exact couples and nilpotent spaces

Abstract: We will introduce spectral sequences via exact couples and outline how to derive the Serre spectral sequence from algebraic topology. Then, we will introduce nilpotent spaces, which are a type of space in many ways dual to a CW complex, before explaining how the Serre spectral sequence can be used to derive some of their properties. For example, the homology groups of a nilpotent space are finitely generated if and only if its homotopy groups are finitely generated.

Algebraic Topology on 15 November 2022 at 16:00

Speaker: Foling Zou (University of Michigan)
Title: Nonabelian Poincare duality theorem in equivariant factorization homology

Abstract: The factorization homology are invariants of n-dimensional manifolds with some fixed tangential structures that take coefficients in suitable En-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group G by monadic bar construction following Kupers--Miller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by Dotto--Moi--Patchkoria--Reeh.

Ergodic Theory and Dynamical Systems on 15 November 2022 at 14:00

Speaker: Mar Giralt (Universitat Politecnica de Catalunya)
Title: Chaotic dynamics, exponentially small phenomena and Celestial Mechanics

Abstract: A fundamental problem in dynamical systems is to prove that a given model has chaotic dynamics. One of the methods employed to prove this type of motions is to verify the existence of transversal intersections between the stable and unstable manifolds of certain objects. Then, there exists a theorem (the Smale-Birkhoff homoclinic theorem) which ensures the existence of chaotic motions. In this talk we present a method to analyze the distance and transversality between certain stable and unstable manifolds when a small perturbation is added to an integrable system. In particular, we consider the case where the distance between manifolds is exponentially small. This implies that this difference cannot be detected by expanding the manifolds into a series with respect to the small perturbation parameter. Therefore, classical perturbation theory cannot be applied. Finally, we apply these techniques to a celestial mechanics problem. In particular, we study the Lagrange point L_3 in the restricted planar circular 3-body problem.

Algebra on 14 November 2022 at 17:00

Speaker: Jay Taylor (University of Manchester)
Title: TBC

Abstract: TBC

Number Theory on 14 November 2022 at 15:00

Speaker: Rosa Winter (KCL)
Title: Density of rational points on del Pezzo surfaces of degree 1

Abstract: Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.

I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

Colloquium on 11 November 2022 at 16:00

Speaker: Yan Fyodorov (King's College London)
Title: "Escaping the crowds": extreme values and outliers in rank-1 non-normal deformations of GUE/CUE

Abstract: Rank-1 non-normal deformations of GUE/CUE provide the simplest model for describing resonances in a quantum chaotic system decaying via a single open channel. In the case of GUE we provide a detailed description of an abrupt restructuring of the resonance density in the complex plane as the function of channel coupling, identify the critical scaling of typical extreme values, and finally describe how an atypically broad resonance (an outlier) emerges from the crowd. In the case of CUE we are further able to study the Extreme Value Statistics of the ''widest resonances'' and find that in the critical regime it is described by a distribution nontrivially interpolating between Gumbel and Frechet. The presentation will be based on the joint works with Boris Khoruzhenko and Mihail Poplavskyi.

Combinatorics on 11 November 2022 at 14:00

Speaker: Adva Mond (University of Cambridge)
Title: Minimum degree edge-disjoint Hamilton cycles in random directed graphs

Abstract: At most how many edge-disjoint Hamilton cycles does a given directed graph contain? A trivial upper bound is the minimum between the minimum out- and in-degrees. We show that a typical random directed graph D(n,p) contains precisely this many edge-disjoint Hamilton cycles, given that p >= (log^3 n)/n, which is optimal up to a factor of log^

Applied Mathematics on 11 November 2022 at 12:00

Speaker: Fabian Spill - POSTPONED TO TERM 2 (Birmingham)
Title: Mechanics, Geometry and Topology of Health and Disease

Abstract: TBAExperimental biologists traditionally study biological functions as well as diseases mostly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. However, many diseases also have a mechanical component which is critical to their deadliness. Notably, cancer kills mostly through metastasis, where the cancer cells acquire the capability to change their physical attachments and migrate. Such mechanical alterations also change geometrical features, such as the cell shape, or topological features, such as the organisation of vascular networks and cellular neighbourhoods within a tissue.

While some of these mechanical, geometrical or topological features in biology are long known, the traditional perspective is to consider them as emergent from molecular features. However, mechanical, geometrical and topological features can also affect the molecular state of a cell. Therefore, the most complete view of many biological systems is to consider them as a complex mechano-chemical systems. Diseases such as cancer are then interpreted as perturbations to this system that cannot be solely explained by considering one feature in isolation (such as a single mutation that ‘causes’ cancer).

I will discuss several examples of systems where this mechanical/geometrical/topological coupling to molecular features plays a crucial role: cells that change their shape, blood vessel cells that open gaps to let cancer cells pass during metastasis, and mitochondria that change their organisation in diabetes.

Mathematics Teaching and Learning on 10 November 2022 at 16:00

Speaker: Edmund Robertson (St. Andrews)
Title: MacTutor – a collection of great mathematicians?

Abstract: In my talk I will look at questions such as: Is MacTutor a collection of great mathematicians? What is a “great mathematician?” How did I choose whom to write about?

Junior Analysis and Probability Seminar on 10 November 2022 at 13:00

Speaker: Simon Gabriel (University of Warwick)
Title: On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations

Abstract: TBA

Probability Seminar on 09 November 2022 at 16:00

Speaker: Kevin Yang (UC Berkeley)
Title: Time-dependent KPZ equation from non-equilibrium Ginzburg-Landau SDEs

Abstract: This talk has two goals. The first is the derivation of a time-dependent KPZ equation (TDKPZ) from a time-inhomogeneous Ginzburg-Landau model. To our knowledge, said TDKPZ has not yet been derived from microscopic considerations. It has a nonlinear twist that is not seen in the usual KPZ equation, making it a more interesting SPDE.

The second goal is the universality of the method (for deriving TDKPZ), which should work beyond Ginzburg-Landau. In particular, we answer a question of deriving (TD)KPZ from asymmetric particle systems under natural fluctuation-scale versions of the assumptions in Yau’s relative entropy method and a log-Sobolev inequality. This gives some progress on open questions posed at a workshop on KPZ at the American Institute of Math. Time permitting, future directions (of both pure and applied mathematical flavors) will be discussed.

Algebraic Geometry on 09 November 2022 at 15:00

Speaker: Beihui Yuan (Swansea)
Title: 16 Betti diagrams of Gorenstein Calabi-Yau varieties and a Betti stratification of Quaternary Quartic Forms

Abstract: Motivated by the question of finding all possible projectively normal Calabi-Yau 3-folds in 7-dimensional projective spaces, we proved that there are 16 possible Betti diagrams for arithmetically Gorenstein ideals with regularity 4 and codimension 4. Among them, 8 Betti diagrams have been identified with those of Calabi-Yau 3-folds appeared in a list of 11 families founded by Coughlan-Golebiowski-Kapustka-Kapustka. Another 8 cannot be Betti diagrams of any smooth irreducible nondegenerate 3-fold. Based on the apolarity correspondence between Gorenstein ideals and homogeneous polynomials, and on our results on 16 Betti diagrams, we describe a stratification of the space of quartic forms in four variables.

This talk is based on the paper “Calabi-Yau threefolds in P^n and Gorenstein rings” by Hal Schenck, Mike Stillman and Beihui Yuan, and the preprint “Quaternary quartic forms and Gorenstein rings” by Michal and Grzegorz Kapustka, Kristian Ranestad, Hal Schenck, Mike Stillman and Beihui Yuan.

Algebraic Topology on 08 November 2022 at 16:00

Speaker: Thibault Décoppet (Oxford)
Title: Fusion 2-Categories associated to 2-groups

Abstract: Motivated by the cobordism hypothesis, which provides a correspondence between fully dualizable objects and fully extended framed TQFTs, it is natural to seek out interesting examples of fully dualizable objects. In dimension four, the fusion 2-categories associated to 2-groups are examples of fully dualizable objects. In my talk, I will begin by reviewing the 2-categorical notion of Cauchy completion, and recall the definition of a fusion 2-category in detail. Then, I will explain how one can construct a fusion 2-category of 2-vector spaces graded by 2-group, and how this construction can be twisted using a 4-cocycle. Finally, it is important to understand when two such fusion 2-categories yield equivalent TQFTs. The answer is provided by the notion of Morita equivalence between fusion 2-categories, which will be illustrated using some examples.

Number Theory on 08 November 2022 at 15:00

Speaker: George Boxer (Imperial)
Title: Higher Hida theory for Siegel modular varieties

Abstract: The goal of higher Hida theory is to study the ordinary part of coherent cohomology of Shimura varieties integrally. We introduce a higher coherent cohomological analog of Hida's space of ordinary p-adic modular forms, which is defined as the ordinary part of the coherent cohomology with "partial compact support" of the ordinary Igusa variety. Then we give an analog of Hida's classicality theorem in this setting. This is joint work with Vincent Pilloni.

Ergodic Theory and Dynamical Systems on 08 November 2022 at 14:00

Speaker: Donald Robertson (University of Manchester)
Title: Dynamical Cubes and Ergodic Theory

Abstract: n recent works Kra, Moreira, Richter and I showed that positive density sets always contain sums of any finite number of infinite sets, and a shift of the self-sum of an infinite set. The main step in our approach was to prove the existence of certain dynamical configurations described via limit points of orbits. In this talk I will describe what these configurations are, and explain how ergodic theory can be used to deduce their existence.

Partial Differential Equations and their Applications on 08 November 2022 at 12:00

Speaker: Angeliki Menegaki (IHES)
Title: TBA

Abstract: TBA

Algebra on 07 November 2022 at 17:00

Speaker: Michael Bate (University of York)
Title: TBC

Abstract: TBC

Colloquium on 04 November 2022 at 16:00

Speaker: Giovanni Alberti (Pisa)
Title: Small sets in Geometric Measure Theory and Analysis

Abstract: Many relevant problems in Geometric Measure Theory can be ultimately understood in terms of the structure of certain classes of sets, which can be loosely described as "small" (in some sense or another). In this talk I will review a few of these problems and related results, and highlight the connections to other areas of Analysis.

Combinatorics on 04 November 2022 at 14:00

Speaker: Alp Müyesser (UCL)
Title: Hypergraphs defined by groups

Abstract: This talk will be about a genre of problems where one looks for spanning structures in hypergraphs where vertices represent group elements, and edges represent solutions to systems of equations. Problems expressible using this framework include the Hall-Paige conjecture, the n-queens problem, the harmonious labelling conjecture, Snevily's subsquare conjecture, and many others. We will discuss an absorption-based attack on problems of this type which has resolved many longstanding conjectures in the area.

Joint work with Alexey Pokrovskiy.

Applied Mathematics on 04 November 2022 at 12:30

Speaker: Emma Davis (Warwick)
Title: Using compartmental ODE models to forecast the elimination of macro-parasitic diseases

Abstract: Standard compartmental models of infectious disease transmission work by categorising a population by stage of infection and then building a system of differential equations that govern the density or number of individuals in each category, e.g. the SIR model has compartments for susceptible (S), infectious (I) and recovered/removed (R) individuals. This makes sense when we are interested in the number of individuals infected over an epidemic where infection is a binary state, as measured by prevalence and incidence, but is less useful for macro-parasitic diseases, where infection is instead classified by the number of macro-parasites inhabiting any given individual. Models for macro-parasitic diseases therefore more commonly consider the number of parasites per individual (their parasite “burden”) or, on a population scale, the mean parasite burden. Common biological features of macro-parasitic diseases, such as sexual reproduction of the parasites or indirect transmission routes, and aggregation between individuals, can result in interesting dynamics at low prevalence, which I will discuss using the example of the macro-parasitic disease lymphatic filariasis.

Applied Mathematics on 04 November 2022 at 12:00

Speaker: Christian Vaquero-Stainer (Warwick)
Title: The sedimentation dynamics of thin, rigid disks

Abstract: Sedimentation problems arise in a wide range of natural and industrial processes and exhibit a rich array of phenomenology. A particular motivation for this study is the size segregation of graphene flakes, for which a dominant method is centrifugation in a viscous fluid (Khan 2012). We present a numerical investigation of the sedimentation dynamics of thin, deformed circular disks sedimenting freely under gravity in an otherwise quiescent, Stokesian fluid. In the first part of this study, we address singularities which arise in the fluid pressure and velocity gradient at the edge of the disk, by developing an augmented finite element method to capture the singularities with analytic functions. In the second part of the study, we deploy this method in a fluid-structure interaction framework to examine the behaviour of two distinct classes of disk shape, namely cylindrically- and conically-deformed disks with one and two planes of symmetry, respectively. We explore the geometry-driven dynamics and the bifurcation structure which arises for the conically-deformed disk as the level of asymmetry is varied.

Geometry and Topology on 03 November 2022 at 14:00

Speaker: Becca Winarski (College of the Holy Cross)
Title: Polynomials, branched covers, and trees

Abstract: Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

Probability Seminar on 02 November 2022 at 16:00

Speaker: Alessandra Cipriani (University College London)
Title: Properties of the gradient squared of the Gaussian free field

Abstract: TBA

Algebraic Geometry on 02 November 2022 at 15:00

Speaker: Michel van Garrel (Birmingham)
Title: Log Mirror Symmetry

Abstract: Start with a smooth Fano variety X and a smooth anticanonical divisor D. Consider the problem of counting maps from the projective line to X that meet D along a curve in only one point. While this problem is intractable directly, in this joint work with Helge Ruddat and Bernd Siebert, we use toric dualities to translate the problem into a dual problem in a dual geometry. There the problem turns into a problem of computing period integrals, which we can readily solved via the techniques of Picard-Fuchs equations.

In my talk, I will limit to the case of X the projective plane and D a smooth conic. The before-mentioned toric dualities are the constructions of the Gross-Siebert programme. I hope to convey the observation that the dualities are natural and that the translation from counting problem to period integral is as well.

Ergodic Theory and Dynamical Systems on 01 November 2022 at 14:00

Speaker: Carlos Matheus (Ecole Polytechnique)
Title: Elliptic dynamics on certain SU(2) and SU(3) character varieties

Abstract: In this talk, we discuss the action of a hyperbolic element of SL(2,Z) on the SU(2) and SU(3) character varieties of once-punctured torii. This is based on a joint work with G. Forni, W. Goldman and S. Lawton.

Algebra on 31 October 2022 at 17:00

Speaker: Sean Eberhard (University of Cambridge)
Title: The Boston--Shalev conjecture for conjugacy classes

Abstract: The Boston--Shalev conjecture (proved by Fulman and Guralnick in 2015) asserts that in any nonabelian simple group G in any nontrivial permutation action the proportion of derangements is at least some absolute constant c > 0. Since the set of derangements is closed under conjugacy it is also natural to ask about the proportion of *conjugacy classes* containing derangements. It is easy to see that this version of the question has a negative answer for alternating groups, but Guralnick and Zalesski asked whether it holds for groups of Lie type. I will outline a proof. We can also (1) extend to the case of almost simple groups, which is not true for the original conjecture, and (2) deduce the original conjecture, which amounts to a simplification of the Fulman--Guralnick proof. The key turns out to be a kind of analytic number theory for palindromic polynomials. This is ongoing work with Daniele Garzoni.

Number Theory on 31 October 2022 at 15:00

Speaker: Yoav Gath (Cambridge)
Title: Lattice point statistics for Cygan–Koranyi balls

Abstract: Euclidean lattice point counting problems, the classical example of which is the Gauss circle problem, are an important topic in classical analysis and have been the driving force behind much of the developments in the area of analytic number theory in the 20th century. In this talk, I will introduce the lattice point counting problem for (2q+1)-dimensional Cygan–Koranyi balls, namely, the problem of establishing error estimates for the number of integer lattice points lying inside Heisenberg dilates of the unit ball with respect to the Cygan–Koranyi norm. I will explain how this problem arises naturally in the context of the Heisenberg groups, and how it relates to the Euclidean case (and in particular to the Gauss circle problem). I will survey some of the major results obtained to date for this lattice point counting problem, and in particular, results related to the fluctuating nature of the error term.

Colloquium on 28 October 2022 at 16:00

Speaker: Tom Gur (Warwick Computer Science)
Title: Quantum algorithms and additive combinatorics

Abstract: I will discuss a new connection between quantum computing and additive combinatorics, which allows for boosting the power of quantum algorithms. Namely, I will show a framework that uses generalisations of Bogolyubov’s lemma and Sander’s quasi-polynomial Bogolyubov-Ruzsa lemma to transform quantum algorithms that are only correct on a small number of inputs into quantum algorithms that are correct on all inputs.

Applied Mathematics on 28 October 2022 at 12:00

Speaker: Maciej Buze (Birmingham)
Title: Mathematical analysis of atomistic fracture and related phenomena in crystalline materials

Abstract: The modelling of atomistic fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial and exciting challenges, both on the theoretical and practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms such as (i) crack propagation; (ii) near-crack tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results. In this talk I will aim to summarise on-going efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will introduce a framework giving rise to well-defined models for which regularity and stability of solutions can be discussed (topic of my PhD thesis at Warwick). I will then show how the theory can be used to set up practical simulations, such as Mode I fracture of silicon on the (111) cleavage plane using state-of-the-art interatomic potentials. I will also outline how this framework can be used to rigorously derive upscaled models of near-crack-tip plasticity. Finally, I will also talk about challenges in addressing the surface effects.

Geometry and Topology on 27 October 2022 at 14:00

Speaker: Daniel Berlyne (University of Bristol)
Title: Braid groups of graphs

Abstract: The braid group of a space X is the fundamental group of its configuration space, which tracks the motion of some number of particles as they travel through X. When X is a graph, the configuration space turns out to be a special cube complex, in the sense of Haglund and Wise. I show how these cube complexes are constructed and use graph of groups decompositions to provide methods for computing braid groups of various graphs, as well as criteria for a graph braid group to split as a free product. This has various applications, such as characterising various forms of hyperbolicity in graph braid groups and determining when a graph braid group is isomorphic to a right-angled Artin group.

Algebraic Geometry on 26 October 2022 at 15:00

Speaker: Arman Sarikyan (Edinburgh)
Title: On the Rationality of Fano-Enriques Threefolds

Abstract: A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

Postgraduate on 26 October 2022 at 12:00

Speaker: Robin Visser (University of Warwick)
Title: Hilbert's Tenth Problem

Abstract: Can you find four distinct positive integers $w,x,y,z$ such that $w^3+x^3=y^3+z^3$ ?

If that's too easy, try finding a non-trivial integer solution to $x^4+y^4+z^4=w^4$.

And good luck finding any integral solution to $x^3+y^3+z^3=114$.

This all begs the question of whether we can construct a general algorithm to determine whether any given Diophantine equation has integer solutions. David Hilbert posed this exact question at the second ICM in 1900, where a negative answer was finally proven 70 years later by Yuri Matiyasevich building on work by Martin Davis, Hilary Putnam and Julia Robinson. In this talk, we'll explore the mathematical ideas behind Hilbert's tenth problem as well as go over many surprising applications, extensions to other number fields, and how this relates to several other famous open problems!

Algebraic Topology on 25 October 2022 at 16:00

Speaker: Severin Bunk (Oxford)
Title: Functorial field theories from differential cocycles

Abstract: In this talk I will demonstrate how differential cocycles give rise to (bordism-type) functorial field theories (FFTs). I will discuss some background on smooth FFTs, differential cohomology and higher gerbes with connection as a geometric model for differential cocycles before explaining the general principle for how to obtain smooth FFTs from higher gerbes. In the second part, I will focus on the two-dimensional case. Here I will present a concrete, geometric construction of two-dimensional smooth FFTs on background manifolds, starting from gerbes with connection. This is related to WZW theories. If time permits, I will comment on an extension of this construction which produces open-closed field theories.

Ergodic Theory and Dynamical Systems on 25 October 2022 at 14:00

Speaker: Maryam Hosseini (Open University)
Title: About Minimal Dynamics on the Cantor Set

Abstract: Dimension group is an operator algebraic object related to minimal dynamical systems on the Cantor set. In this talk after a quick review of some definitions of dimension group, the {\it topological and algebraic rank} of Cantor minimal systems are considered and we will see how the rank of a Cantor system is dominated by the rank of its extensions.

Partial Differential Equations and their Applications on 25 October 2022 at 12:00

Speaker: Alexandra Tzella (University of Birmingham)
Title: TBA

Abstract: TBA

Algebra on 24 October 2022 at 17:00

Speaker: Alexandre Zalesski (UEA)
Title: Some problems on representations of simple algebraic groups

Abstract: Some open questions on the weight structure of tensor-decomposable representations of simple algebraic groups will be discussed.

Number Theory on 24 October 2022 at 15:00

Speaker: Aleksander Horawa (Oxford)
Title: Motivic action on coherent cohomology of Hilbert modular varieties

Abstract: A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

Colloquium on 21 October 2022 at 16:00

Speaker: Pierre Raphael (Cambridge)
Title: Singularity formation for super critical waves

Abstract: Give a wave packet an initial energy and let it propagate in the whole space, then in the linear regime, the wave packet will scatter. But in non linear regimes, part of the energy may concentrate to form coherent non linear structures which propagate without deformation (solitons). And in more extreme cases singularities may form. Whether or not singular structures arise is a delicate problem which has attracted a considerable amount of works in both mathematics and physics, in particular in the super critical regime which is the heart of the 6Th Clay problem on singularity formation for three dimensional viscous incompressible fluids. For another classical model like the defocusing Non Linear Schrodinger equation (NLS), Bourgain ruled out in a breakthrough work (1994) the existence of singularities in the critical case, and conjectured that this should extend to the super critical one. I will explain how the recent series of joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris 6) shades a new light on super critical singularities: in fact there exist super critical singularities for (NLS), and the new underlying mechanism is directly connected to the first description of singularities for three dimensional viscous compressible fluids.

Combinatorics on 21 October 2022 at 14:00

Speaker: Daniel Iľkovič (Masaryk University)
Title: Quasirandom tournaments

Abstract: A directed graph H is quasirandom-forcing for tournaments if the limit (homomorphic) density of H in a sequence of tournaments is 2^−|E(H)| if and only if the sequence is quasirandom. The cyclic orientation of a cycle of length k is quasirandom-forcing if and only if k = 2 mod 4. We study a generalization of this result: what orientations of a cycle of length k are quasirandom-forcing? We show that no orientation of an odd cycle is quasirandom-forcing and classify which orientations of even cycles of length up to 10 are quasirandom-forcing. This is joint work with Andrzej Grzesik, Bartosz Kielak and Dan Kráľ

Applied Mathematics on 21 October 2022 at 12:00

Speaker: Francis Aznaran (Oxford)
Title: Finite element methods for the Stokes–Onsager–Stefan–Maxwell equations of multicomponent flow

Abstract: The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. This represents some of the first rigorous numerics for the coupling of multicomponent molecular diffusion with compressible convective flow. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons. This is joint work with Alexander Van-Brunt.

Geometry and Topology on 20 October 2022 at 14:00

Speaker: Henry Bradford (Cambridge)
Title: TBA

Abstract: TBA

Algebraic Geometry on 19 October 2022 at 15:00

Speaker: Nivedita Viswanathan (Loughborough)
Title: On the Rationality of Fano-Enriques Threefolds

Abstract: There has been a lot of development recently in understanding the existence of Kahler-Einstein metrics on Fano manifolds due to the Yau-Tian-Donaldson conjecture, which gives us a way of looking at this problem in terms of the notion of K-stability. In particular, this problem is solved in totality for smooth del Pezzo surfaces by Tian. For del Pezzo surfaces with quotient singularities, there are partial results. In this talk, we will consider singular del Pezzo surfaces which are quasi-smooth, well-formed hypersurfaces in weighted projective space, and understand what we can say about their K-stability. This is ongoing joint work with In-Kyun Kim and Joonyeong Won.

Postgraduate on 19 October 2022 at 12:00

Speaker: Ruzhen Yang (University of Warwick)
Title: Beilinson spectral sequence and its reverse problems on PP^2

Abstract: Derived category is widely accepted as the natural environment to study homological algebra. We will study the structure of the bounded derived category of coherent sheaves on projective space via the semi-orthogonal decomposition (based on the Beilinson's theorem) and comparison (by a theorem by A. Bondal). As an example, we will give explicit free resolutions of some sheaves on P2 using the Beilinson spectral sequence. We will also discuss the reverse problem where we give a condition to when the complex given by the spectral sequence is a resolution of the ideal sheaf of three points.

Algebraic Topology on 18 October 2022 at 16:00

Speaker: Thomas Read (Warwick)
Title: G-typical Witt vectors with coefficients and the norm

Abstract: The norm is an important construction on equivariant spectra, most famously playing a key role in the work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem. Witt vectors are an algebraic construction first used in Galois theory in the 1930s, but later finding applications in stable equivariant homotopy theory. I will describe a new generalisation of Witt vectors that can be used to compute the zeroth equivariant stable homotopy groups of the norm $N_e^G Z$, for $G$ a finite group and $Z$ a connective spectrum.

Ergodic Theory and Dynamical Systems on 18 October 2022 at 14:00

Speaker: Joe Thomas (Durham University)
Title: Poisson statistics, short geodesics and small eigenvalues on hyperbolic punctured spheres

Abstract: For hyperbolic surfaces, there is a deep connection between the geometry of closed geodesics and their spectral theoretic properties. In this talk, I will discuss recent work with Will Hide (Durham), where we study both sides of this relationship for hyperbolic punctured spheres. In particular, we consider Weil-Petersson random surfaces and demonstrate Poisson statistics for counting functions of closed geodesics with lengths on scales 1/sqrt(number of cusps), in the large cusp regime. Using similar ideas, we show that typical hyperbolic punctured spheres with many cusps have lots of arbitrarily small eigenvalues. Throughout, I will contrast these findings to the setting of closed hyperbolic surfaces in the large genus regime.

Algebra on 17 October 2022 at 17:00

Speaker: Kamilla Rekvényi (Imperial College London)
Title: The Orbital Diameter of Primitive Permutation Groups

Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on ΩxΩ component wise. Define the orbitals to be the orbits of G on ΩxΩ. The diagonal orbital is the orbital of the form ∆ = {(α, α)|α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α,β)∈ Γ with α,β∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs. There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups.

Number Theory on 17 October 2022 at 15:00

Speaker: Maria Rosaria Pati (Padova)
Title: L-invariants for cohomological representations of PGL(2) over an arbitrary number field

Abstract: In this talk I will construct the automorphic L-invariant attached to a cuspidal representation π of PGL(2) over an arbitrary number field F, and a prime p of F such that the local component πp is the Steinberg representation and π is non-critical at p. I will show that, if F is totally real then the automorphic L-invariant attached to π and p agrees with the derivatives of the Up-eigenvalue of the p-adic family passing through π. From this I will deduce the equality between the automorphic L-invariant and the Fontaine-Mazur L-invariant of the associated Galois representation. This is a joint work with Lennart Gehrmann.

Colloquium on 14 October 2022 at 16:00

Speaker: David Rand (Warwick)
Title: An appreciation of Christopher Zeeman

Abstract: In writing a biographical memoir of Zeeman for the Royal Society, I appreciated even more what a remarkable character he was, both in terms of his life, his leadership, his mathematics and his breadth of interests. I discovered a number of aspects that I don't think are very well known about his life and his contributions to topology, catastrophe theory and our department. In this colloquium I will try and give an overview of this.

Combinatorics on 14 October 2022 at 14:00

Speaker: Candy Bowtell (University of Warwick)
Title: The n-queens problem

Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n). In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption. This is joint work with Peter Keevash.

Applied Mathematics on 14 October 2022 at 12:00

Speaker: Philip Herbert (Heriot-Watt)
Title: Shape optimisation with Lipschitz functions

Abstract: In this talk, we discuss a novel method in PDE constrained shape optimisation. We begin by introducing the concept of PDE constrained shape optimisation. While it is known that many shape optimisation problems have a solution, their approximation in a meaningful way is non-trivial. To find a minimiser, it is typical to use first order methods. The novel method we propose is to deform the shape with fields which are a direction of steepest descent in the topology of W^1_\infty. We present an analysis of this in a discrete setting along with the existence of directions of steepest descent. Several numerical experiments will be considered which compare a classical Hilbertian approach to this novel approach.

Geometry and Topology on 13 October 2022 at 14:00

Speaker: Claudio Llosa Isenrich (KIT)
Title: Finiteness properties, subgroups of hyperbolic groups and complex hyperbolic lattices

Abstract: Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely man cells of dimension at most n, generalising finite presentability, which is equivalent to type $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. We use methods from complex geometry to show that every uniform arithmetic lattice with positive first Betti number in $PU(n,1)$ admits a finite index subgroup, which maps onto the integers with kernel of type $F_{n-1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.

Algebraic Topology on 11 October 2022 at 16:00

Speaker: Sebastian Chenery (University of Southampton)
Title: On Pushout-Pullback Fibrations

Abstract: We will discuss recent work inspired by a paper of Jeffrey and Selick, where they ask whether the pullback bundle over a connected sum can itself be homeomorphic to a connected sum. We provide a framework to tackle this question through classical homotopy theory, before pivoting to rational homotopy theory to give an answer after taking based loop spaces.

Ergodic Theory and Dynamical Systems on 11 October 2022 at 14:00

Speaker: Sabrina Kombrink (University of Birmingham)
Title: TBA

Abstract: TBA

Algebra on 10 October 2022 at 17:00

Speaker: Gareth Tracey (University of Warwick)
Title: Primitive amalgams and the Goldschmidt-Sims conjecture

Abstract: The Classification of Finite Simple Groups has led to substantial progress on deriving sharp order bounds in various natural families of finite groups. One of the most well-known instances of this is Sims' conjecture, which states that a point stabiliser in a primitive permutation group has order bounded in terms of its smallest non-trivial orbit length (this was proved by Cameron, Praeger, Saxl and Seitz using the CFSG in 1983). In the meantime, Goldschmidt observed that a generalised version of Sims' conjecture, which we now call the \emph{Goldschmidt--Sims conjecture}, would lead to important applications in graph theory. In this talk, we will describe the conjecture, and discuss some recent progress. Joint work with L. Pyber.

Number Theory on 10 October 2022 at 15:00

Speaker: Lambert A'Campo (Oxford)
Title: Galois representations and cohomology of congruence subgroups

Abstract: In this talk I will explain what it means to attach Galois representations to the cohomology of arithmetic locally symmetric spaces arising from congruence subgroups. In the case of GL(2) over imaginary CM fields (the method also works for GL(n)) I will explain how to prove, under certain conditions, that the Galois representations constructed by Harris–Lan–Taylor–Thorne and Scholze have good p-adic Hodge theoretic properties.

DAGGER on 10 October 2022 at 14:00

Speaker: Marco Linton (University of Oxford)
Title: Poison subgroups for hyperbolic groups

Abstract: It is a well-known result that hyperbolic groups cannot contain certain `poison' subgroups. A lot of progress has been made towards understanding when the converse to this statement also holds. This includes several positive results, but also several negative results. In this talk, I will introduce hyperbolic groups, discuss some of these results and present the current state of the art for the class of one-relator groups.

Colloquium on 07 October 2022 at 16:00

Speaker: Michela Ottobre (Heriot-Watt)
Title: Interacting Particle systems and (Stochastic) Partial Differential equations: modelling, analysis and computation

Abstract: The study of Interacting Particle Systems (IPSs) and related kinetic equations has attracted the interest of the mathematics and physics communities for decades. Such interest is kept alive by the continuous successes of this framework in modelling a vast range of phenomena, in diverse fields such as biology, social sciences, control engineering, economics, game theory, statistical sampling and simulation, neural networks etc. While such a large body of research has undoubtedly produced significant progress over the years, many important questions in this field remain open. We will (partially) survey some of the main research directions in this field and discuss open problems.

Combinatorics on 07 October 2022 at 14:00

Speaker: Mustazee Rahman (University of Durham)
Title: Suboptimality of local algorithms for optimization on sparse graphs

Abstract: Suppose we want to find the largest independent set or maximal cut in a large yet sparse graph, where the average vertex degree is constant. These are two basic optimization problems relevant to both theory and practice. For typical, or rather random sparse graphs, many algorithms proceed by way of local decision rules. Examples include Glauber dynamics, Belief propagation, etc. I will explain a form of local algorithm that captures many of these. I will then explain how they fail to find optimal independent sets or cuts once the average degree of the graph gets large. Along the way, we will find connections to entropy and spin glasses.

Geometry and Topology on 06 October 2022 at 14:00

Speaker: Grace Garden (University of Sydney)
Title: Earthquakes on the once-punctured torus

Abstract: We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in hyperbolic geometry, the second representation theory. The two methods align, providing both a geometric and an algebraic interpretation of the earthquake deformations. Pictures are given for earthquakes across multiple coordinate systems for Teichmüller space. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Algebra on 05 October 2022 at 16:17

Speaker: Matija Vidmar (University of Ljubljana)
Title: Noise Boolean algebras: classicality, blackness and spectral independence

Abstract: Informally speaking, a noise Boolean algebra is an aggregate of pieces of information, subject to statistical independence properties relative to an underlying notion of chance. More formally, it is a distributive sublattice of the lattice of all sub-sigma-fields of a given probability space, each element of which admits an independent complement. A noise Boolean algebra is classical (resp. black) when all its random variables are stable (resp. sensitive) under infinitesimal perturbations of its basic ingredients. For instance, the Wiener and Poisson noises are classical, but certain noises of percolation and coalescence are black. We shall see that classicality and blackness are respectively characterized by existence and non-existence of certain so-called spectral independence probabilities that we shall introduce. Associated preprint: https://drive.google.com/file/d/1cLOHpHG_xgqPYmsbVIQmmYx08pqk4H6m/view

Postgraduate on 05 October 2022 at 12:00

Speaker: Sunny Sood (University of Warwick)
Title: Homological stability for $O_{n,n}$

Abstract: Motivated by Hermitian K-Theory, we study the homological stability of the split orthogonal group $O_{n,n}$. Specifically, let $R$ be a commutative local ring with infinite residue field such that $2 \in R^{*}$. We prove that the natural homomorphism $H_{k}(O_{n,n}(R) ; \mathbb{Z}) \rightarrow H_{k}(O_{n+1,n+1}(R); \mathbb{Z})$ is an isomorphism for $k \leq n-1$ and surjective for $k \leq n$. This will be an excellent opportunity to introduce esoteric concepts such as group homology and hyperhomology spectral sequences at the postgraduate seminar. This is all joint work with my supervisor Dr Marco Schlichting.

Partial Differential Equations and their Applications on 04 October 2022 at 12:00

Speaker: Tobias Barker (University of Bath)
Title: A quantitative approach to the Navier–Stokes equations

Abstract: ecently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

Number Theory on 03 October 2022 at 15:00

Speaker: Matteo Tamiozzo (Warwick)
Title: Perfectoid quaternionic Shimura varieties and the Jacquet–Langlands correspondence

Abstract: The Hodge–Tate period map can be thought of as a p-adic analogue of the Borel embedding. However, unlike its complex counterpart, it is not injective, and the pushforward of the constant sheaf via the Hodge–Tate period map encodes interesting arithmetic information. In the setting of quaternionic Shimura varieties, I will explain the relation between the structure of this complex of sheaves and level raising and the Jacquet–Langlands correspondence. I will then discuss applications to the study of the cohomology of quaternionic Shimura varieties. I will illustrate most of the arguments in the simplest setting of modular and Shimura curves. This is joint work with Ana Caraiani.

DAGGER on 03 October 2022 at 14:00

Speaker: Aleksi Pyörälä (University of Oulu)
Title: Normal numbers in self-conformal sets

Abstract: During recent years, the prevalence of normal numbers in natural subsets of the reals has been an active research topic in fractal geometry. The general idea is that in the absence of any special arithmetic structure, almost all numbers in a given set should be normal, in every base. In our recent joint work with Balázs Bárány, Antti Käenmäki and Meng Wu we verify this for self-conformal sets on the line. The result is a corollary of a uniform scaling property of self-conformal measures: roughly speaking, a measure is said to be uniformly scaling if the sequence of successive magnifications of the measure equidistributes, at almost every point, for a common distribution supported on the space of measures. Dynamical properties of these distributions often give information on the geometry of the uniformly scaling measure.